# The negative association property for the absolute values of random
variables equidistributed on a generalized Orlicz ball

Marcin Pilipczuk<EMAIL_ADDRESS> Jakub Onufry Wojtaszczyk
<EMAIL_ADDRESS>
Department of Mathematics, Computer Science and Mechanics
University of Warsaw
ul. Banacha 2, 02-097 Warsaw, Poland
Partially supported by MEiN Grant no 1 PO3A 012 29

###### Abstract

Random variables equidistributed on convex bodies have received quite a lot of
attention in the last few years. In this paper we prove the negative
association property (which generalizes the subindependence of coordinate
slabs) for generalized Orlicz balls. This allows us to give a strong
concentration property, along with a few moment comparison inequalities. Also,
the theory of negatively associated variables is being developed in its own
right, which allows us to hope more results will be available.

Moreover, a simpler proof of a more general result for $\ell_{p}^{n}$ balls is
given.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation
    2. 1.2 Results
    3. 1.3 Motivations
    4. 1.4 Acknowledgements
  2. 2 Easy facts
    1. 2.1 Simplifying
    2. 2.2 Simple proportion lemmas
  3. 3 The $\ell_{p}^{n}$ ball case
  4. 4 The generalized Orlicz ball case — preliminaries, the proper measure, lens sets
    1. 4.1 Idea of the proof
    2. 4.2 Definitions
    3. 4.3 The generalized Orlicz ball lemmas
    4. 4.4 $1/m$-concave functions and proper measures
    5. 4.5 Lens sets and $\Theta$ functions
  5. 5 The $\Theta$ theorem
    1. 5.1 Preparations for divisibility
    2. 5.2 Almost horizontal divisions
    3. 5.3 ${\varepsilon}$-appropriateness of lens sets
  6. 6 The transfinite induction
    1. 6.1 Starting the transfinite induction
    2. 6.2 The induction step for successor ordinals
    3. 6.3 The induction step for limit ordinals
  7. 7 $\Theta$ functions on Orlicz balls
    1. 7.1 The one-dimensional case — the $\phi$ functions
    2. 7.2 The general case — the $\psi$ function

##  1 Introduction

###  1.1 Notation

We shall begin by introducing the notation used throughout the paper. For any
set $A$ by ${\mathbf{1}}_{A}$ we shall denote the characteristic function of
$A$. As usually, ${\mathbb{R}}$ and ${\mathbb{R}}_{+}$ will denote the reals
and the non-negative reals respectively. By ${\mathbb{R}}^{k}$ we shall mean
the $k$-dimensional Euclidean space equipped with the standard scalar product
$\left\langle\cdot,\cdot\right\rangle$, the Lebesgue measure denoted by
$\lambda$ or $\lambda_{k}$ and a system of orthonormal coordinates
$x_{1},x_{2},\ldots,x_{k}$. By ${\mathbb{R}}_{+}^{k}$ we mean the generalized
positive quadrant, that is the set
$\\{(x_{1},\ldots,x_{k})\in{\mathbb{R}}^{k}:\forall_{i}\ x_{i}\geq 0\\}$. For
a given set $K\subset{\mathbb{R}}^{k}$ by $K_{+}$ we shall denote the positive
quadrant of $K$, that is $K\cap{\mathbb{R}}_{+}^{k}$. For a given set $A$ by
${\bar{A}}$ we will denote the complement of $A$.

For a measure $\mu$ on ${\mathbb{R}}^{n}$ and an affine subspace
$H\subset{\mathbb{R}}^{n}$, by the projection of $\mu$ onto $H$ we mean the
measure $\mu_{H}$ defined by $\mu_{H}(C)=\mu(\\{x\in{\mathbb{R}}^{n}:P(x)\in
C\\})$, where $P$ is the orthogonal projection onto $H$. If $\mu$ is given by
a density function $m$ and $K\subset H\subset{\mathbb{R}}^{n}$, then by the
restriction of $\mu$ to $K$ we mean the measure $\mu_{|K}$ on $H$ given with
the density $m\cdot{\mathbf{1}}_{K}$. By the support of a function
$m:X\to{\mathbb{R}}$, denoted ${\rm supp}m$, we mean ${\rm cl}\\{x\in
X:m(x)\neq 0\\}$. If $\mu$ is a measure, then by ${\rm supp}\mu$ we mean the
smallest closed set $A$ such that $\mu(\bar{A})=0$. In the cases we consider,
when $\mu$ will be given by a density $m$, we will always have $supp\mu={\rm
supp}m$.

We shall call a set $K\subset{\mathbb{R}}^{n}$ a symmetric body if it is
convex, bounded, central-symmetric (i.e. if $x\in K$ then $-x\in K$) and has a
non-empty interior. A body $K\subset{\mathbb{R}}^{n}$ is called 1-symmetric if
for any $({\varepsilon}_{1},\ldots,{\varepsilon}_{n})\in\\{-1,1\\}^{n}$ and
any $(x_{1},\ldots,x_{n})\in K$ we have
$({\varepsilon}_{1}x_{1},\ldots,{\varepsilon}_{n}x_{n})\in K$. Such a body is
sometimes called unconditional.

A function $f:{\mathbb{R}}_{+}{\rightarrow}{\mathbb{R}}_{+}\cup\\{\infty\\}$
is called a Young function if it is convex, $f(0)=0$ and $\exists_{x}:f(x)\neq
0$, $\exists_{x\neq 0}:f(x)\neq\infty$. If we have $n$ Young functions
$f_{1},\ldots,f_{n}$, then the set

$K=\\{(x_{1},\ldots,x_{n}):\sum_{i=1}^{n}f_{i}(|x_{i}|)\leq 1\\}$

is a 1-symmetric body in ${\mathbb{R}}^{n}$. Such a set is called a
generalized Orlicz ball, also known in the literature as a modular sequence
space ball.

We shall call a Young function $f$ proper if it does not attain the $+\infty$
value and $f(x)>0$ for $x>0$. A generalized Orlicz ball is called proper if it
can be defined by proper Young functions.

If the coordinates of the space ${\mathbb{R}}^{n}$ are denoted
$x_{1},x_{2},\ldots,x_{n}$, the appropriate Young functions will be denoted
$f_{1},f_{2},\ldots,f_{n}$, with the assumption $f_{i}$ is applied to $x_{i}$.
If some of the coordinates are denoted $x,y,z,\ldots$, the appropriate Young
functions will be denoted $f_{x},f_{y},f_{z},\ldots$, with the assumption that
$f_{x}$ is applied to $x$, $f_{y}$ to $y$ and so on.

A function $f:{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is called increasing
(decreasing) if $x\geq y$ implies $f(x)\geq f(y)$ ($f(x)\leq f(y)$) — we do
not require a sharp inequality. A function
$f:{\mathbb{R}}^{k}{\rightarrow}{\mathbb{R}}$ or
$f:{\mathbb{R}}_{+}^{k}{\rightarrow}{\mathbb{R}}$ is called coordinate-wise
increasing (decreasing), if for $x_{i}\geq y_{i}$, $i=1,2,\ldots,n$ we have
$f(x_{1},\ldots,x_{k})\geq f(y_{1},\ldots,y_{n})$ ($f(x_{1},\ldots,x_{k})\leq
f(y_{1},\ldots,y_{n})$). A set $A\subset{\mathbb{R}}_{+}^{k}$ is called a
c-set, if for $x_{i}\geq y_{i}\geq 0$, $i=1,2,\ldots,n$ and
$(x_{1},\ldots,x_{n})\in A$ we have $(y_{1},\ldots,y_{n})\in A$. For a
coordinate-wise increasing function
$f:{\mathbb{R}}_{+}^{k}{\rightarrow}{\mathbb{R}}$ the sets
$f^{-1}((-\infty,t])$ are c-sets, and conversely the characteristic function
of a c-set is a coordinate-wise decreasing function on ${\mathbb{R}}_{+}^{k}$.
Similarily a function $f:{\mathbb{R}}_{+}^{k}{\rightarrow}{\mathbb{R}}$ is
radius-wise increasing if $f(tx_{1},tx_{2},\ldots,tx_{n})\geq
f(x_{1},x_{2},\ldots,x_{n})$ for $t>1$, and a set $A$ is a radius-set if its
characteristic function is radius-wise decreasing.

We say a function $f:{\mathbb{R}}^{n}{\rightarrow}{\mathbb{R}}_{+}$ is log-
concave if $\ln f$ is concave. A measure $\mu$ on ${\mathbb{R}}^{n}$ is called
log-concave if for any nonempty $A,B\subset{\mathbb{R}}^{n}$ and $t\in(0,1)$
we have $\mu(tA+(1-t)B)\geq\mu(A)^{t}\mu(B)^{1-t}$. A classic theorem by
Borell (see [Bo74]) states that any log-concave density not concentrated on
any affine hyperplane has a density function, and that function is log-
concave. A random vector in ${\mathbb{R}}^{n}$ is said to be log-concave if
its distribution is log-concave.

A sequence of random variables $(X_{1},\ldots,X_{n})$ is said to be negatively
associated, if for any coordinate-wise increasing bounded functions $f,g$ and
disjoint sets $\\{i_{1},\ldots,i_{k}\\}$ and
$\\{j_{1},\ldots,j_{l}\\}\subset\\{1,\ldots,n\\}$ we have

${\rm
Cov}\big{(}f(X_{i_{1}},\ldots,X_{i_{k}}),g(X_{j_{1}},\ldots,X_{j_{l}})\big{)}\leq
0.$ (1.1.1)

We say that the sequence $(X_{j})$ is weakly negatively associated if
inequality (1.1.1) holds for $l=1$, and very weakly negatively associated if
(1.1.1) holds for $l=k=1$.

For a 1-symmetric body $K\subset{\mathbb{R}}^{n}$ we can treat the body, or
its positive quadrant, as a probability space, with the normalized Lebesgue
measure as the probability. Formally, we consider $\Omega=K$, the Borel
subsets of $K$ as the $\sigma$-family and
${\mathbb{P}}=\frac{1}{\lambda(K)}\lambda$ as the probability measure. We do
similarly for $K_{+}$. We also define $n$ random variables
$X_{1},\ldots,X_{n}$, with $X_{i}$ being the $i$-th coordinate of a point
$\omega\in K_{+}$ or $K$.

###  1.2 Results

Our main subject of interest is to prove negative associacion type properties
for some classes of symmetric bodies in ${\mathbb{R}}^{n}$. An straightforward
approach is bound to fail due to the following proposition:

######  Proposition 1.1.

If for a 1-symmetric body $K$ we consider the random vectors uniformly
distributed on $K$ (not just on $K_{+}$) and the coordinate variables are very
weakly negatively associated, then they are pairwise independent, and thus $K$
is a rescaled cube.

###### Proof.

Take any $i,j\in\\{1,\ldots,n\\}$ and any increasing functions
$f,g:{\mathbb{R}}{\rightarrow}{\mathbb{R}}$. Then $f^{\circ}(x)=-f(-x)$ is
increasing too. $K$ is 1-symmetric, so $(X_{i},X_{j})$ has the same joint
distribution as $(-X_{i},X_{j})$, so

${\rm Cov}(f^{\circ}(X_{i}),g(X_{j}))={\rm
Cov}\big{(}-f(-X_{i}),g(X_{j}))=-{\rm Cov}(f(X_{i}),g(X_{j})\big{)}.$

If both ${\rm Cov}(f(X_{i}),g(X_{j}))$ and $-{\rm Cov}(f(X_{i}),g(X_{j}))$ are
non-positive, then ${\rm Cov}(f(X_{i}),g(X_{j}))=0$. This holds for every
$i,j,f,g$. In particular for every $a,b$ we have

${\mathbb{P}}(X_{i}\in[a,\infty)\cap
X_{j}\in[b,\infty))-{\mathbb{P}}(X_{i}\in[a,\infty))\cdot{\mathbb{P}}(X_{j}\in[b,\infty))={\rm
Cov}({\mathbf{1}}_{[a,\infty)},{\mathbf{1}}_{[b,\infty)})=0.$

A standard argument shows that $X_{i}$ and $X_{j}$ are independent, thus the
density of ${\mathbf{1}}_{K}$ is a product density, so $K$ has to be a product
of intervals. ∎

Thus, even very weak negative associacion for coordinate variables occurs only
in the trivial case. The problem becomes more interesting if we look at the
variables $|X_{i}|$ (or, equivalently, restrict ourselves to $X_{i}\geq 0$).

K. Ball and I. Perissinaki in [BP98] prove the subindependence of coordinate
slabs for $\ell^{p}$ balls, from which very weak negative association of
$(|X_{1}|,\ldots,|X_{n}|)$ is a simple consequence. In the paper [W06]
Corollary 3.2 states that the sequence of variables $(|X_{1}|,\ldots,|X_{n}|)$
is very weakly negatively associated for generalized Orlicz balls.

In this paper we shall prove that for a generalized Orlicz ball the sequence
of variables $(|X_{1}|,\ldots,|X_{n}|)$ is negatively associated:

######  Theorem 1.2.

Let $K$ be an generalized Orlicz ball, and let $X_{i}$ be the coordinates of a
random vector uniformly distributed on $K$. Then the sequence $|X_{i}|$ is
negatively associated.

We shall also prove an even stronger property of $\ell_{p}^{n}$ balls:

######  Theorem 1.3.

Take any $p\in[1,\infty)$ and any $n\in{\mathbb{N}}$. Let
$m:{\mathbb{R}}_{+}{\rightarrow}{\mathbb{R}}_{+}$ be any log-concave function
and let $\mu$ be the measure on ${\mathbb{R}}^{n}$ with the density at $x$
equal to $m(\|x\|_{p}^{p})$ normalized to be a probability measure. Let
$I=\\{i_{1},\ldots,i_{k}\\},J=\\{j_{1},\ldots,j_{l}\\}$ be two disjoint
subsets of $\\{1,2,\ldots,n\\}$, and let
$f:{\mathbb{R}}_{+}^{k}{\rightarrow}{\mathbb{R}}$,
$g:{\mathbb{R}}_{+}^{l}{\rightarrow}{\mathbb{R}}$ be any radius-wise
increasing functions bounded on ${\rm supp}\mu$. Let
$X=(X_{1},X_{2},\ldots,X_{n})$ be the vector distributed according to $\mu$.
Then

${\rm
Cov}(f(|X_{i_{1}}|,|X_{i_{2}}|,\ldots,|X_{i_{k}}|),g(|X_{j_{1}}|,|X_{j_{2}}|,\ldots,|X_{j_{l}}|))\leq
0.$

This is an equivalent of the above theorem, but the uniform distribution is
replaced by the class of distribution with the density being a log-concave
function of the $p$-th power of the $p$-th norm, and the coordinate-wise
increasing function replaced by radius-wise decreasing functions.

Let us comment on the organization of the paper. In the following subsection
we shall state the main results and show a few corollaries which motivate
these results. Section 2 is a collection of general lemmas, which allow us to
reformulate the problem in a simpler fashion. In Section 3 a simple proof for
the $\ell_{p}^{n}$ result is given. Section 4 introduces the definitions used
in dealing with the generalized Orlicz ball case and investigates the basic
properties of the defined objects. Section 5 states the $\theta$-theorem,
which is the main tool of the proof, and gives a part of the proof. Section 6
contains the second part of the proof, which is a large transfinite inductive
construction. Finally Section 7 applies the $\theta$-theorem to obtain the
result for generalized Orlicz balls.

###  1.3 Motivations

This study was motivated by a desire to link the results achieved in convex
geometry in [ABP03] for $\ell_{p}$ balls and in [W06] for generalized Orlicz
balls with an established theory, which will hopefully allow us to avoid
repeating proofs already made in a more general case. For example, a form of
the Central Limit Theorem for negative associated variables was already known
in 1984 (see [N84]). We also hope some new observations can be made using this
approach.

The negative association property is stronger then the sub-independence of
coordinate slabs, which has been studied in the context of the Central Limit
Theorem (see [ABP03], [BP98]). The statement of Theorem 1.3 was motivated by
Theorem 6 of [BGMN05], where a proof of subindependence of coordinate slabs is
given for a different class of measures with density dependent on the $p$-th
norm, also including the uniform measure and the normalized cone measure on
the surface.

An example that can prove useful for applications in convex geometry is a pair
of comparison inequalities due to Shao (see [S00]). First, notice that as
$|X_{i}|$ are negatively associated, they remain negatively associated when
multiplied by any non-negative scalars (which amounts to multiplying $X_{i}$
by any scalars) and after the addition of any constant scalars. Thus the
vectors $|a_{i}X_{i}|-c_{i}$ are negatively associated for any
$a_{i},c_{i}\in{\mathbb{R}}$. Shao’s inequalities, when applied to our case it
will state the following:

######  Theorem 1.4.

Let $K\subset{\mathbb{R}}^{n}$ be a generalized Orlicz ball,
$(a_{i})_{i=1}^{n}$ be any sequence of reals and $(X_{i})_{i=1}^{n}$ be the
coordinates of the random vector uniformly distributed on $K$. Then for any
convex function $f:{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ we have

${\mathbb{E}}f\Big{(}\sum_{i=1}^{n}|a_{i}X_{i}|\Big{)}\leq{\mathbb{E}}f\Big{(}\sum_{i=1}^{n}|a_{i}X_{i}^{\star}|\Big{)},$

where $X_{i}^{\star}$ denote independent random variables with $X_{i}$ and
$X_{i}^{\star}$ having the same distribution for each $i$. Additionally, if
$f$ is increasing, then for any sequence of reals $(c_{i})_{i=1}^{n}$ we have

${\mathbb{E}}f\Big{(}\max_{k=1,2,\ldots,n}\sum_{i=1}^{k}|a_{i}X_{i}|-c_{i}\Big{)}\leq{\mathbb{E}}f\Big{(}\max_{k=1,2,\ldots,n}\sum_{i=1}^{k}|a_{i}X_{i}^{\star}|-c_{i}\Big{)}.$

A more direct consequence is a moment comparision theorem suggested by R.
Latała (note we compare the moments of the sums of variables, and not their
absolute values):

######  Theorem 1.5.

Let $K\subset{\mathbb{R}}^{n}$ be a generalized Orlicz ball,
$(a_{i})_{i=1}^{n}$ be a sequence of reals and $(X_{i})_{i=1}^{n}$ be the
coordinates of the random vector uniformly distributed on $K$. Then for any
even positive integer $p$ we have

${\mathbb{E}}\Big{(}\sum_{i=1}^{n}a_{i}X_{i}\Big{)}^{p}\leq{\mathbb{E}}\Big{(}\sum_{i=1}^{n}a_{i}X_{i}^{\star}\Big{)}^{p},$

with $X_{i}^{\star}$ defined as before.

###### Proof.

When we open the brackets in $(\sum a_{i}X_{i})^{p}$ the summands in which at
least one $X_{i}$ appears with an odd exponent average out to zero, as $K$ is
1-symmetric. Thus what is left is a sum of elements of the form

$(a_{i}X_{1})^{2\alpha_{1}}(a_{2}X_{2})^{2\alpha_{2}}\ldots(a_{n}X_{n})^{2\alpha_{n}}=|a_{1}X_{1}|^{2\alpha_{1}}|a_{2}X_{2}|^{2\alpha_{2}}\ldots|a_{n}X_{n}|^{2\alpha_{n}}.$

If we put $f(a_{1}x_{1})=(a_{1}x_{1})^{2\alpha_{1}}$ and
$g(a_{2}x_{2},\ldots,a_{n}x_{n})=(a_{2}x_{2})^{2\alpha_{2}}\cdot\ldots\cdot(a_{n}x_{n})^{2\alpha_{n}}$,
applying negative association we get

$\displaystyle{\mathbb{E}}|a_{1}X_{1}|^{2\alpha_{1}}|a_{2}X_{2}|^{2\alpha_{2}}\ldots|a_{n}X_{n}|^{2\alpha_{n}}$
$\displaystyle\leq{\mathbb{E}}|a_{1}X_{1}|^{2\alpha_{1}}{\mathbb{E}}|a_{2}X_{2}|^{2\alpha_{2}}\ldots|a_{n}X_{n}|^{2\alpha_{n}}$
$\displaystyle={\mathbb{E}}|a_{1}X_{1}^{\star}|^{2\alpha_{1}}{\mathbb{E}}|a_{2}X_{2}|^{2\alpha_{2}}\ldots|a_{n}X_{n}|^{2\alpha_{n}}$
$\displaystyle={\mathbb{E}}|a_{1}X_{1}^{\star}|^{2\alpha_{1}}|a_{2}X_{2}|^{2\alpha_{2}}\ldots|a_{n}X_{n}|^{2\alpha_{n}}.$

Repeating this process inductively we separate all the variables and get

$\displaystyle{\mathbb{E}}\Big{(}\sum_{i=1}^{n}a_{i}X_{i}\Big{)}^{p}$
$\displaystyle=\sum_{\alpha_{1}+\ldots+\alpha_{n}=p/2}C_{\alpha_{1},\ldots,\alpha_{n}}{\mathbb{E}}|a_{1}X_{1}|^{2\alpha_{1}}|a_{2}X_{2}|^{2\alpha_{2}}\ldots|a_{n}X_{n}|^{2\alpha_{n}}\leq$
$\displaystyle\leq\sum_{\alpha_{1}+\ldots+\alpha_{n}=p/2}C_{\alpha_{1},\ldots,\alpha_{n}}{\mathbb{E}}|a_{1}X_{1}^{\star}|^{2\alpha_{1}}|a_{2}X_{2}^{\star}|^{2\alpha_{2}}\ldots|a_{n}X_{n}^{\star}|^{2\alpha_{n}}=$
$\displaystyle={\mathbb{E}}\Big{(}\sum_{i=1}^{n}a_{i}X_{i}^{\star}\Big{)}^{p}.$

∎

Finally, we can apply Shao’s maximal inequality to get a exponential
concentration of the euclidean norm. Theorem 3 in [S00] states:

######  Theorem 1.6.

Let $(X_{i})_{i=1}^{n}$ be a sequence of negatively associated random
variables with zero means and finite second moments. Let
$S_{k}=\sum_{i=1}^{k}X_{i}$ and $B_{n}=\sum_{i=1}^{n}{\mathbb{E}}X_{i}^{2}$.
Then for all $x>0$, $a>0$ and $0<\alpha<1$

${\mathbb{P}}\Big{(}\max_{1\leq k\leq n}|S_{k}|\geq x\Big{)}\leq
2{\mathbb{P}}(\max_{1\leq k\leq
n}|X_{k}|>a)+\frac{2}{1-\alpha}\exp\Bigg{(}-\frac{x^{2}\alpha}{2(ax+B_{n})}\cdot\Big{(}1+\frac{2}{3}\ln\Big{(}1+\frac{ax}{B_{n}}\Big{)}\Big{)}\Bigg{)}.$

We say $K\subset{\mathbb{R}}^{n}$ is in isotropic position if
$\lambda_{n}(K)=1$ and ${\mathbb{E}}X_{i}^{2}=L_{K}^{2}$ for some constant
$L_{K}$ (any bounded convex set with a non-empty interior can be moved into
isotropic position by an affine transformation, for more on this subject see
e.g. [MS86]). Notice that if $|X_{i}|$ are negatively associated and $f_{i}$
are increasing, then $f_{i}(|X_{i}|)$ are also negatively associated. Thus the
sequence $(X_{i}^{2}-L_{K}^{2})_{i=1}^{n}$ for $X=(X_{i})_{i=1}^{n}$ uniformly
distributed on a generalized Orlicz ball is also negatively associated. The
moments of log-concave variables are comparable (see for instance [KLO96],
Section 2, remark 5), thus we have

${\mathbb{E}}(X_{i}^{2}-L_{K}^{2})^{2}={\mathbb{E}}X_{i}^{4}+L_{K}^{4}-2L_{K}^{2}{\mathbb{E}}X_{i}^{2}={\mathbb{E}}X_{i}^{4}-L_{K}^{4}\leq
5L_{K}^{4}.$

If we put $\alpha=1/2$ and $x=nt$ in Shao’s inequality and apply the bound we
got above for the variance we get

######  Corollary 1.7.

Let $K\subset{\mathbb{R}}^{n}$ be a generalized Orlicz ball in isotropic
position, and $(X_{i})_{i=1}^{n}$ be the coordinates of the random vector
uniformly distributed on $K$. Then for any $t>0$, $a>0$ we have:

$\displaystyle{\mathbb{P}}\Big{(}\max_{1\leq k\leq
n}\Big{|}\sum_{i=1}^{k}(X_{i}^{2}-L_{K}^{2})\Big{|}>nt\Big{)}$
$\displaystyle\leq 2{\mathbb{P}}\Big{(}\max_{1\leq k\leq
n}|X_{k}^{2}-L_{K}^{2}|>a\Big{)}+$
$\displaystyle+4\exp\Bigg{(}-\frac{nt^{2}}{4(at+5L_{K}^{4})}\cdot\bigg{(}1+\frac{2}{3}\ln\Big{(}1+\frac{at}{5L_{K}^{4}}\Big{)}\bigg{)}\Bigg{)}.$

To apply this result probably an idea on what order of convergence is possible
to achieve with this formula would be needed. To this end we give the
following corollary:

######  Corollary 1.8.

Let $K\subset{\mathbb{R}}^{n}$ be a generalized Orlicz ball in isotropic
position, and $(X_{i})_{i=1}^{n}$ be the coordinates of the random vector
uniformly distributed on $K$. Then for any $t>0$ we have:

${\mathbb{P}}\Big{(}\Big{|}\frac{\sum_{i=1}^{n}X_{i}^{2}}{n}-L_{K}^{2}\Big{|}>t\Big{)}\leq
Ce^{-cnt^{2}}+Cne^{-c\sqrt[3]{nt}},$

where $C$ and $c$ are universal constants independent of $t$, $n$ and $K$.

For $t>t_{0}$ a better bound (of the order of $e^{-t\sqrt{n}}$) is due to
Bobkov and Nazarov (see [BN03]). However, frequently a bound for
$t{\rightarrow}0$ is needed — for instance the proof of the Central Limit
Theorem for convex bodies uses bounds for the concentration of the second norm
for small $t$ (see for instance [ABP03]). In full generality (ie. for an
arbitrary log-concave isotropic measure and for arbitrary $t$) such a result
is given in a very recent paper by Klartag (see [K07]) with worse exponents —
the bound for the probabilty is of the order of $e^{t^{3.33}n^{0.33}}$.
Previous proofs of such results (see [FGP07], [K07,2]) gave a logarithmic
dependence of the exponent on $n$. The bound given in the corollary above is
very rough, and in any particular case it is very likely it may be improved.
However, we give it in order to show an explicit exponential bound in the
concentration inequality which is uniform for all generalized Orlicz balls in
a given dimension and applies for any $t>0$.

###### Proof.

Obviously

${\mathbb{P}}\Bigg{(}\Bigg{|}\frac{\sum_{i=1}^{n}X_{i}^{2}}{n}-L_{K}^{2}\Bigg{|}>t\Bigg{)}\leq{\mathbb{P}}\Bigg{(}\max_{1\leq
k\leq n}\Bigg{|}\sum_{i=1}^{k}(X_{i}^{2}-L_{K}^{2})\Bigg{|}>nt\Bigg{)},$

so we have only to bound the right hand side in Corollary 1.7. Put
$a=\sqrt[3]{n^{2}t^{2}}$. We know (see [MP89]) that $L_{K}^{2}$ is bounded by
some universal constant $L$ independent of $n$ and $K$ for any 1-symmetric
body in ${\mathbb{R}}^{n}$. If $c$ is small enough and $C$ large enough, then
for $a<L_{K}^{2}$ we have

$Cne^{-c\sqrt[3]{nt}}=Cne^{-c\sqrt{a}}\geq 1.$

Thus we may consider only the case $a>L_{K}^{2}$.

In this case

$\displaystyle{\mathbb{P}}(\max_{1\leq k\leq n}|X_{k}^{2}-L_{K}^{2}|>a)$
$\displaystyle\leq n\max_{1\leq k\leq
n}{\mathbb{P}}(|X_{k}^{2}-L_{K}^{2}|>a)=n\max_{k}{\mathbb{P}}(X_{k}^{2}>a+L_{K}^{2})$
$\displaystyle\leq
n\max_{k}{\mathbb{P}}(X_{k}^{2}>a)=n\max_{k}{\mathbb{P}}(|X_{k}|>\sqrt{a}).$

Due to the Brunn-Minkowski inequality $X_{k}$ is log-concave (see for instance
[Ga02]), we know that $Var(X_{k})\leq L_{K}^{2}<C$ and ${\mathbb{E}}X_{k}=0$,
and thus $P(|X_{k}|>t)\leq c_{1}e^{-c_{2}t}$ for some universal constants
$c_{1}$ and $c_{2}$ independent of the distribution of $X_{k}$ and of $t$
(Borell’s Lemma, see for instance [MS86]). Thus we get

${\mathbb{P}}\Big{(}\max_{1\leq k\leq n}|X_{k}^{2}-L_{K}^{2}|>a\Big{)}\leq
c_{1}e^{-c_{2}\sqrt{a}}=c_{1}e^{-c_{2}\sqrt[3]{nt}}.$

In the second part we shall simply bound

$\Bigg{(}1+\frac{2}{3}\ln\Big{(}1+\frac{at}{5L_{K}^{4}}\Big{)}\Bigg{)}\geq 1.$

Then

$4\exp\Bigg{(}-\frac{nt^{2}}{4(at+5L_{K}^{4})}\cdot\bigg{(}1+\frac{2}{3}\ln\Big{(}1+\frac{at}{5L_{K}^{4}}\Big{)}\bigg{)}\Bigg{)}\leq
4\exp\big{(}-\frac{nt^{2}}{4n^{2/3}t^{5/3}+20L_{K}^{4}}\big{)}\leq
Ce^{-c\sqrt[3]{nt}}+Ce^{-cnt^{2}}.$

∎

###  1.4 Acknowledgements

We would very much like to thank Rafał Latała, who encouraged us to write the
paper, was the first person to read it and check the reasoning, and helped
improve the paper in innumerable aspects. He also taught us most of what we
know in the subject.

We would also like to thank prof. Stanisław Kwapień, who first suggested to us
the idea of searching for negative-association type properties for convex
bodies.

##  2 Easy facts

###  2.1 Simplifying

We want to prove inequality (1.1.1) for various classes of functions
(coordinate-wise increasing in the case of Theorem 1.2 and radius-wise
increasing in the case of Theorem 1.3). We may assume $k+l=n$ by putting
$\tilde{g}(x_{j_{1}},\ldots,x_{j_{l}},x_{r_{1}},\ldots,x_{r_{n-l-k}})=g(x_{j_{1}},\ldots,x_{j_{l}})$.
For convienience we shall assume that the Lebesgue volume of $K_{+}$ is 1
(inequality (1.1.1) is invariant under homothety). It will be more convienient
to work with c-sets or radius-sets than with functions, which motivates the
following Lemma:

######  Lemma 2.1.

Let $\mu$ be any probability measure on ${\mathbb{R}}_{+}^{n}$ and let
$X=(X_{1},X_{2},\ldots,X_{n})$ be the random vector distributed according to
$\mu$. Assume that for given $0\leq k,l\leq n$ we have two families of bounded
functions $\mathcal{F}$ on ${\mathbb{R}}_{+}^{k}$ and $\mathcal{G}$ on
${\mathbb{R}}_{+}^{l}$. Let
$\mathcal{A}=\\{f^{-1}(-\infty,t]:f\in\mathcal{F},t\in{\mathbb{R}}\\}$, and
similarly $\mathcal{B}$ for $\mathcal{G}$. If for any $A\in\mathcal{A}$ and
$B\in\mathcal{B}$ we have

$\mu(A\times
B)\mu({\bar{A}}\times{\bar{B}})\leq\mu(A\times{\bar{B}})\mu({\bar{A}}\times
B),$ (2.1.1)

then inequality (1.1.1) holds for $X$ and any
$f\in\mathcal{F},g\in\mathcal{G}$.

In particular, if inequality (2.1.1) holds for any $k$ and for any c-sets
$A,B$, then the random variables $X_{1},X_{2},\ldots,X_{n}$ are negatively
associated.

###### Proof.

Let us take any two functions $\mathcal{F}\ni
f:{\mathbb{R}}_{+}^{k}{\rightarrow}{\mathbb{R}}$ and $\mathcal{G}\ni
g:{\mathbb{R}}_{+}^{l}{\rightarrow}{\mathbb{R}}$. As covariance is bilinear
and is 0 if one of the functions is constant, we may assume without loss of
generality that $f$ and $g$ are non-negative. For non-negative functions we
have

$f(x)=\int_{0}^{\infty}{\mathbf{1}}_{f^{-1}[t,\infty)}(x)\ dt.$

Thus (again, by the bilinearity of the covariance) we can restrict ourselves
to functions $f$ and $g$ of the form $1-{\mathbf{1}}_{A}$ and
$1-{\mathbf{1}}_{B}$, where $A\in\mathcal{A}$ and $B\in\mathcal{B}$. Since
${\rm Cov}(1-{\mathbf{1}}_{A},1-{\mathbf{1}}_{B})={\rm
Cov}({\mathbf{1}}_{A},{\mathbf{1}}_{B})$, we have to prove that ${\rm
Cov}({\mathbf{1}}_{A},{\mathbf{1}}_{B})\leq 0$.

Let us denote by $\mathbf{X}$ the $k$-dimensional vector
$(X_{i_{1}},\ldots,X_{i_{k}})$ on which $f$ is taken, and by $\mathbf{Y}$ the
$l$-dimensional vector on which $g$ is taken. Then

$\displaystyle{\rm
Cov}\big{(}{\mathbf{1}}_{A}({\mathbf{X}}),{\mathbf{1}}_{B}({\mathbf{Y}})\big{)}$
$\displaystyle={\mathbb{E}}{\mathbf{1}}_{A}({\mathbf{X}}){\mathbf{1}}_{B}({\mathbf{Y}})-{\mathbb{E}}{\mathbf{1}}_{A}({\mathbf{X}}){\mathbb{E}}{\mathbf{1}}_{B}({\mathbf{Y}})=\mu(A\times
B)-\mu(A\times{\mathbb{R}}^{l})\mu({\mathbb{R}}^{k}\times B)$
$\displaystyle=\mu(A\times
B)\mu\big{(}(A\cup{\bar{A}})\times(B\cup{\bar{B}})\big{)}-\mu\big{(}A\times(B\cup{\bar{B}})\big{)}\mu\big{(}(A\cup{\bar{A}})\times
B\big{)}$ $\displaystyle=\mu(A\times
B)\mu({\bar{A}}\times{\bar{B}})-\mu(A\times{\bar{B}})\mu({\bar{A}}\times B),$

which is non-positive by (2.1.1). ∎

###  2.2 Simple proportion lemmas

During the course of further proofs we shall frequently need to compare two
ratios of integrals of the same functions over different sets.

In this subsection we will demonstrate some simple properties of ratios of
integrals.

######  Fact 2.2.

Let $a,b\geq 0$ and $c,d>0$. Then the following are equivalent:

  * •

$\frac{a}{c}\geq\frac{b}{d}$,

  * •

$\frac{a}{c}\geq\frac{a+b}{c+d}$,

  * •

$\frac{a+b}{c+d}\geq\frac{b}{d}$.

Whenever there is equality in one of the inequalities, all aforementioned
fractions are equal.

######  Lemma 2.3.

Let $\mu$ be a non-negative measure on ${\mathbb{R}}$ supported on the
(possibly unbounded) interval $[l_{\mu},r_{\mu}]$. Suppose that
$f,g,h:{\mathbb{R}}{\rightarrow}{\mathbb{R}}_{+}$ are functions bounded on
${\rm supp}\mu$, positive on the interior of their supports, satisfying:

  1. 1.

The support of any function $u\in\\{f,g,h\\}$ is an interval $[l_{u},r_{u}]$
(possibly unbounded),

  2. 2.

$\frac{f}{g}$ is a decreasing function where defined, and $r_{f}\leq r_{g}$,

  3. 3.

$h$ is an increasing function,

Then:

  1. (1a)

For any $a<b<c$, $b\in(l_{\mu},r_{\mu})\cap(l_{g},r_{g})$ we have

$\frac{\int_{a}^{b}f(x)d\mu}{\int_{a}^{b}g(x)d\mu}\geq\frac{f(b)}{g(b)}\hbox{
and }\frac{f(b)}{g(b)}\geq\frac{\int_{b}^{c}f(x)d\mu}{\int_{b}^{c}g(x)d\mu}$

whenever both sides of an inequality are defined.

  2. (1b)

Moreover, if for some $a<b<c$ we have two equalities in inequality (1a) then
$\frac{f(x)}{g(x)}$ is constant on $(a,c)\cap{\rm supp}g\cap{\rm supp}\mu$ and
for any $a\leq s<t\leq c$

$\frac{\int_{s}^{t}f(x)d\mu}{\int_{s}^{t}g(x)d\mu}$

is equal to $f(b)/g(b)$ if defined.

  3. (2a)

For any points $a,b,c,d$ satisfying $a<b\leq d$ and $a\leq c<d$ we have:

$\frac{\int_{a}^{b}f(x)d\mu}{\int_{a}^{b}g(x)d\mu}\geq\frac{\int_{c}^{d}f(x)d\mu}{\int_{c}^{d}g(x)d\mu}$

whenever both sides are defined.

  4. (2b)

Moreover, if this inequality is an equality and either
$\int_{a}^{c}g(x)d\mu(x)$ or $\int_{b}^{d}g(x)d\mu(x)$ is strictly positive,
then $\frac{f}{g}$ is constant on $[a,d]$ where defined, and we have an
equality for any $a\leq a^{\prime}\leq b^{\prime}\leq d^{\prime}\leq d$ and
$c^{\prime}\in[a^{\prime},d^{\prime}]$ if both sides are defined.

  5. (3)

If $l_{g}=l_{f}$ the following inequality occurs for any interval $I$:

$\frac{\int_{I}f(x)d\mu(x)}{\int_{I}g(x)d\mu(x)}\geq\frac{\int_{I}f(x)h(x)d\mu(x)}{\int_{I}g(x)h(x)d\mu(x)}$

if both sides are defined.

###### Proof.

  1. (1a)

Consider the first inequality. Let $a^{\prime}=\max\\{l_{\mu},l_{g},a\\}$ . We
have $a\leq a^{\prime}<b$ (otherwise the denominator of the left-hand side
would be undefined). Also
$\int_{a}^{b}g(x)d\mu(x)=\int_{a^{\prime}}^{b}g(x)d\mu(x)>0$ and $g>0$ on
$(a^{\prime},b]$ (it has to be positive in $b$ or the right-hand side would be
undefined). Thus

$\frac{\int_{a}^{b}f(x)d\mu(x)}{\int_{a}^{b}g(x)d\mu(x)}\geq\frac{\int_{a^{\prime}}^{b}f(x)}{\int_{a^{\prime}}^{b}g(x)}=\frac{\int_{a^{\prime}}^{b}g(x)\frac{f(x)}{g(x)}}{\int_{a^{\prime}}^{b}g(x)}\geq\frac{\int_{a^{\prime}}^{b}g(x)\frac{f(b)}{g(b)}}{\int_{a^{\prime}}^{b}g(x)}=\frac{f(b)}{g(b)},$

A similar reasoning with $c^{\prime}=\min\\{r_{\mu},r_{g},c\\}$ proves the
second inequality (note $r_{f}\leq r_{g}$, so the first inequality in the
reasoning above becomes an equality).

  2. (1b)

If equality occurs, then $\frac{f(x)}{g(x)}=\frac{f(b)}{g(b)}$ for almost all
$x\in(a^{\prime},c^{\prime})$ as $g$ is strictly positive on
$(a^{\prime},c^{\prime})$. As $\frac{f}{g}$ is decreasing, if it is constant
on almost whole $(a^{\prime},c^{\prime})$, it is constant on the whole
interval and thus

$\frac{\int_{s}^{t}f(x)d\mu(x)}{\int_{s}^{t}g(s)d\mu(x)}=\frac{f(b)}{g(b)}$

if defined for any $s,t\in(a^{\prime},c^{\prime})$. We know
$\int_{a}^{a^{\prime}}g(x)d\mu(x)=\int_{c^{\prime}}^{c}g(x)d\mu(x)=0$, so to
have equalities we also have to have
$\int_{a}^{a^{\prime}}f(x)d\mu(x)=\int_{c^{\prime}}^{c}f(x)d\mu(x)=0$, thus
$\int_{s}^{t}f(x)d\mu(x)=\int_{(s,t)\cap(a^{\prime},c^{\prime})}f(x)d\mu(x)$
and similarly for $g$, thus the thesis.

  3. (2a)

Let $F(x,y)=\int_{x}^{y}f(t)$ and $G(x,y)=\int_{x}^{y}g(t)$. As the left-hand
side is defined, $G(a,b)>0$ and thus $G(a,d)>0$. We apply (1a) to get:

$\frac{F(a,b)}{G(a,b)}\geq\frac{F(b,d)}{G(b,d)}$ (2.2.1)

if the right-hand side is defined and from Fact 2.2 we have

$\frac{F(a,b)}{G(a,b)}\geq\frac{F(a,b)+F(b,d)}{G(a,b)+G(b,d)}=\frac{F(a,d)}{G(a,d)}.$

If the right-hand side in (2.2.1) was not defined, $G(b,d)=0$ and thus
$F(b,d)=0$ as $r_{f}\leq r_{g}$, so
$\frac{F(a,b)}{G(a,b)}\geq\frac{F(a,d)}{G(a,d)}$. Similarly from (1a)

$\frac{F(a,c)}{G(a,c)}\geq\frac{F(c,d)}{G(c,d)}$

if the left-hand side is defined, and thus from Fact 2.2

$\frac{F(a,d)}{G(a,d)}\geq\frac{F(c,d)}{G(c,d)}.$

If the left-hand side was undefined, $G(a,d)=G(c,d)$ and obviously $F(a,d)\geq
F(c,d)$, so we get the same inequality. Linking the two inequalities we get
the thesis.

  4. (2b)

Suppose $G(b,d)>0$. As

$\frac{F(a,b)}{G(a,b)}\geq\frac{F(a,d)}{G(a,d)}\geq\frac{F(c,d)}{G(c,d)}$

and the first and last expressions are equal, all inequalities are in fact
equalities. Thus from the first one of them and Fact 2.2 we get

$\frac{F(a,b)}{G(a,b)}=\frac{F(b,d)}{G(b,d)},$

and applying (1b) we get the thesis.

  5. (3)

Let $I^{\prime}=I\cap{\rm supp}g$. As ${\rm supp}f\subset{\rm supp}g$ all
integrals in the thesis over $I$ are equal to the appropriate integrals over
$I^{\prime}$. Consider the functions $h$ and $\frac{f}{g}$ on the interval
${\rm Int}I^{\prime}$ (note $\frac{f}{g}$ is defined on ${\rm Int}I^{\prime}$)
taken with a measure with density
$\frac{g(x)}{\int_{I^{\prime}}g(t)d\mu(t)}d\mu$ (this is defined as the left-
hand side in the thesis was defined, so $\int_{I^{\prime}}g(t)d\mu(t)>0$).
From the continuous Chebyshev sum inequality (that is, if $F$ is increasing
and $G$ is decreasing, then $\int F\int G\geq\int FG\int 1$) we know

$\displaystyle\int_{I^{\prime}}h(x)\frac{g(x)}{\int_{I^{\prime}}g(t)d\mu(t)}d\mu(x)$
$\displaystyle\int_{I^{\prime}}\frac{f(x)}{g(x)}\frac{g(x)}{\int_{I^{\prime}}g(t)d\mu(t)}d\mu(x)$
$\displaystyle\geq$
$\displaystyle\int_{I^{\prime}}h(x)\frac{f(x)}{g(x)}\frac{g(x)}{\int_{I^{\prime}}g(t)d\mu(t)}d\mu(x)\int_{I^{\prime}}\frac{g(x)}{\int_{I^{\prime}}g(t)d\mu(t)}d\mu(x).$

Multiplying both sides by $[\int_{I^{\prime}}g(t)d\mu(t)]^{2}$ we get the
thesis.

∎

######  Lemma 2.4.

Let $\mu$ be a non-negative measure on $I\subset{\mathbb{R}}$. Suppose
$f,g,p,q:I{\rightarrow}{\mathbb{R}}_{+}$ are functions satisfying
$f(x)g(y)\geq f(y)g(x)$ for $x\geq y$ and $p(x)q(y)\leq p(y)q(x)$ for $x\geq
y$. Then

$\int_{I}p(x)f(x)d\mu(x)\int_{I}q(x)g(x)d\mu(x)\leq\int_{I}p(x)g(x)d\mu(x)\int_{I}q(x)f(x)d\mu(x).$

###### Proof.

Using Fubini’s theorem we have to prove

$\int_{I}\int_{I}p(x)f(x)q(y)g(y)\ d\mu(y)\
d\mu(x)\leq\int_{I}\int_{I}p(y)f(x)q(x)g(y)\ d\mu(y)\ d\mu(x).$

Multiplying sides by two and changing names $x$ and $y$:

$\int_{I}\int_{I}\big{[}p(x)f(x)q(y)g(y)+p(y)f(y)q(x)g(x)-p(x)f(y)q(y)g(x)-p(y)f(x)q(x)g(y)\big{]}\
d\mu(y)\ d\mu(x)\leq 0$
$\int_{I}\int_{I}\big{(}p(x)q(y)-p(y)q(x)\big{)}\big{(}f(x)g(y)-f(y)g(x)\big{)}\
d\mu(y)\ d\mu(x)\leq 0,$

which follows from the assumptions, as the integrand is always non-positive. ∎

######  Lemma 2.5.

Suppose $f,g:X{\rightarrow}{\mathbb{R}}_{+}$ are defined on any set $X$ with a
measure $\mu$. Let $\\{D_{i}\\}_{i\in I}$ be a family of disjoint subsets of
$X$. If

$t\int_{D_{i}}g(x)d\mu(x)\geq\int_{D_{i}}f(x)d\mu(x)\geq
s\int_{D_{i}}g(x)d\mu(x)$

for some $t,s\in{\mathbb{R}}\cup\\{-\infty,\infty\\}$, then

$t\int_{\bigcup_{i}D_{i}}g(x)d\mu(x)\geq\int_{\bigcup_{i}D_{i}}f(x)d\mu(x)\geq
s\int_{\bigcup_{i}D_{i}}g(x)d\mu(x).$

If $X=X_{1}\times X_{2}$ and $\mu=\mu_{1}\otimes\mu_{2}$, and for some set
$D\subset X_{1}\times X_{2}$ and any $x_{1}\in X_{1}$ we have

$t\int_{(\\{x_{1}\\}\times X_{2})\cap
D}g(x)d\mu_{2}(x)\geq\int_{(\\{x_{1}\\}\times X_{2})\cap D}f(x)d\mu_{2}(x)\geq
s\int_{(\\{x_{1}\\}\times X_{2})\cap D}g(x)d\mu_{2}(x),$

then

$t\int_{D}g(x)d\mu(x)\geq\int_{D}f(x)d\mu(x)\geq s\int_{D}g(x)d\mu(x).$

###### Proof.

In the first case, we should add all the inequalities by sides. In the second
case, we should not sum but integrate using Fubini’s theorem. ∎

##  3 The $\ell_{p}^{n}$ ball case

First we shall give the proof for $\ell_{p}^{n}$ balls. Recall the
$\ell_{p}^{n}$ ball is the generalized Orlicz ball defined by the Young
functions $f_{i}(x)=|x|^{p}$. We include this case for two reasons: first, it
is much simpler than the Orlicz ball case, and serves as a good illustration
of what is happening, and second, because we are able to achieve a stronger
result, namely prove Theorem 1.3.

Note that in particular we can take $m$ to be $c_{r}{\mathbf{1}}_{[0,r]}$ to
get the result for the uniform measure on the $\ell_{p}^{n}$ ball. As any
coordinate-wise increasing function is radius-wise increasing, this result is
stronger than the negative associacion property we prove for generalized
Orlicz balls. By a simple approximation argument we can also get the result
above for $\mu$ being the cone measure on the surface of $\ell_{p}^{n}$.

###### Proof.

Let $B_{p}^{n}$ denote the $\ell_{p}^{n}$ ball. Let
$M(x_{1},x_{2},\ldots,x_{n})=(|x_{1}|,|x_{2}|,\ldots,|x_{n}|)$ and let
$\tilde{\mu}$ be defined by $\tilde{\mu}(A)=\mu(M^{-1}(A))$. Notice
$\tilde{\mu}$ describes the distribution of
$(|X_{1}|,|X_{2}|,\ldots,|X_{n}|)$. As $\mu$ is 1-symmetric, we may
equivalently define $\tilde{\mu}$ as $2^{n}$ times the restriction of $\mu$ to
${\mathbb{R}}_{+}^{n}$.

Recall that the cone measure on $\partial B_{p}^{n}$ (that is, the boundary of
$B_{p}^{n}$), which we shall denote $\nu$, is defined for $A\subset\partial
B_{p}^{n}$ by

$\nu_{n}(A)=\frac{\lambda_{n}(ta:t\in{\mathbb{R}},a\in A,ta\in
B_{p}^{n})}{\lambda_{n}(B_{p}^{n})}.$

For this measure we have the polar integration formula:

$\int_{{\mathbb{R}}^{n}}f(x)dx=n\lambda_{n}(B_{p}^{n})\int_{R_{+}}r^{n-1}\int_{\partial
B_{p}^{n}}f(r\theta)d\nu_{n}(\theta)dr.$

Let $C_{n}=n\lambda_{n}(B_{p}^{n})$.

Due to Lemma 2.1 we only need to prove inequality $\tilde{\mu}(A\times
B)\tilde{\mu}({\bar{A}}\times{\bar{B}})\leq\tilde{\mu}(A\times{\bar{B}})\tilde{\mu}({\bar{A}}\times
B)$ for any radius-sets $A,B$, which is equivalent to $\mu(A\times
B)\mu({\bar{A}}\times{\bar{B}})\leq\mu(A\times{\bar{B}})\mu({\bar{A}}\times
B)$. We have:

$\displaystyle\mu(A\times B)$
$\displaystyle=\int_{{\mathbb{R}}^{k}}\int_{{\mathbb{R}}^{n-k}}{\mathbf{1}}_{A}(x){\mathbf{1}}_{B}(y)m(\|x\|_{p}^{p}+\|y\|_{p}^{p})dxdy=$
$\displaystyle=\int_{{\mathbb{R}}_{+}}\int_{\partial
B_{p}^{k}}\int_{{\mathbb{R}}^{n-k}}C_{k}r^{k-1}{\mathbf{1}}_{A}(r\theta){\mathbf{1}}_{B}(y)m(r^{p}+\|y\|_{p}^{p})d\nu_{k}(\theta)drdy$
$\displaystyle=\int_{{\mathbb{R}}_{+}}\bigg{[}\int_{{\mathbb{R}}^{n-k}}{\mathbf{1}}_{B}(y)m(r^{p}+\|y\|_{p}^{p})dy\bigg{]}\bigg{[}\int_{\partial
B_{p}^{k}}{\mathbf{1}}_{A}(r\theta)d\nu_{k}(\theta)\bigg{]}C_{k}r^{k-1}dr.$

Denote
$f_{B}(r)=\int_{{\mathbb{R}}^{n-k}}{\mathbf{1}}_{B}(y)m(r^{p}+\|y\|_{p}^{p})dy$
and $g_{A}(r)=\int_{\partial
B_{p}^{k}}{\mathbf{1}}_{A}(r\theta)d\nu_{k}(\theta)$. Let $\sigma_{1}$ be the
measure on ${\mathbb{R}}_{+}$ with density $C_{k}r^{k-1}$. We can perform
similar operations for the other three expressions in inequality (2.1.1). What
we have to prove becomes the inequality

$\int_{{\mathbb{R}}_{+}}f_{B}(r)g_{A}(r)d\sigma_{1}(r)\int_{{\mathbb{R}}_{+}}f_{{\bar{B}}}(r)g_{{\bar{A}}}(r)d\sigma_{1}(r)\leq\int_{{\mathbb{R}}_{+}}f_{{\bar{B}}}(r)g_{A}(r)d\sigma_{1}(r)\int_{{\mathbb{R}}_{+}}f_{B}(r)g_{{\bar{A}}}(r)d\sigma_{1}(r).$

Due to lemma 2.4 it is enough to prove the following two inequalities:

$\displaystyle f_{B}(r_{1})f_{{\bar{B}}}(r_{2})\geq
f_{B}(r_{2})f_{{\bar{B}}}(r_{1})\hbox{ for }r_{1}\geq r_{2},$ (3.0.1)
$\displaystyle g_{A}(r_{1})g_{{\bar{A}}}(r_{2})\leq
g_{A}(r_{2})g_{{\bar{A}}}(r_{1})\hbox{ for }r_{1}\geq r_{2}.$ (3.0.2)

Inequality (3.0.2) is simple — ${\mathbf{1}}_{A}(r\theta)$ is decreasing as a
function of $r$ for any fixed $\theta$, while
${\mathbf{1}}_{{\bar{A}}}(r\theta)$ is increasing, as $A$ is a radius-set.
Thus $g_{A}(r)$ is decreasing, $g_{{\bar{A}}}$ is increasing, so
$g_{A}(r_{1})\leq g_{A}(r_{2})$ and $g_{{\bar{A}}}(r_{2})\leq
g_{{\bar{A}}}(r_{1})$.

Inequality (3.0.1) will require a bit more work. We have:

$\displaystyle f_{B}(r_{1})$
$\displaystyle=\int_{{\mathbb{R}}^{n-k}}{\mathbf{1}}_{B}(y)m(r_{1}^{p}+\|y\|_{p}^{p})dy$
$\displaystyle=\int_{{\mathbb{R}}_{+}}\int_{\partial
B_{p}^{n-k}}C_{n-k}s^{n-k-1}{\mathbf{1}}_{B}(s\xi)m(r_{1}^{p}+s^{p})d\nu_{n-k}(\xi)dr$
$\displaystyle=\int_{{\mathbb{R}}_{+}}\bigg{[}m(r_{1}^{p}+s^{p})\bigg{]}\bigg{[}\int_{\partial
B_{p}^{n-k}}{\mathbf{1}}_{B}(s\xi)d\nu_{n-k}(\xi)\bigg{]}C_{n-k}s^{n-k-1}ds.$

We are going to use Lemma 2.4 once again. Let
$p_{r_{1}}(s)=m(r_{1}^{p}+s^{p})$ and $q_{B}(s)=\int_{\partial
B_{p}^{n-k}}{\mathbf{1}}_{B}(s\xi)d\nu_{n-k}(\xi)$ and $\sigma_{2}$ the
measure with density $C_{n-k}s^{n-k-1}$. We do the similar calculation for the
other three expressions in inequality (3.0.1), and it becomes

$\int_{{\mathbb{R}}_{+}}p_{r_{1}}(s)q_{B}(s)d\sigma_{2}(s)\int_{{\mathbb{R}}_{+}}p_{r_{2}}(s)q_{{\bar{B}}}(s)d\sigma_{2}(s)\geq\int_{{\mathbb{R}}_{+}}p_{r_{2}}(s)q_{B}(s)d\sigma_{2}(s)\int_{{\mathbb{R}}_{+}}p_{r_{1}}(s)q_{{\bar{B}}}(s)d\sigma_{2}(s).$

Applying Lemma 2.4 we have to prove

$\displaystyle p_{r_{1}}(s_{1})p_{r_{2}}(s_{2})\leq
p_{r_{2}}(s_{1})p_{r_{1}}(s_{2})\hbox{ for }s_{1}\geq s_{2},$ (3.0.3)
$\displaystyle q_{B}(s_{1})q_{{\bar{B}}}(s_{2})\leq
q_{{\bar{B}}}(s_{1})q_{B}(s_{2})\hbox{ for }s_{1}\geq s_{2}.$ (3.0.4)

Inequality (3.0.4) is proved in the same way as inequality (3.0.2) — $q_{B}$
is decreasing and $q_{{\bar{B}}}$ is increasing. Inequality (3.0.3) means

$m(r_{1}^{p}+s_{1}^{p})m(r_{2}^{p}+s_{2}^{p})\leq
m(r_{2}^{p}+s_{1}^{p})m(r_{1}^{p}+s_{2}^{p}),$

which follows from the log-concavity of $m$. ∎

As we saw, this proof was quite simple. Unfortunately, it takes advantage of
the fact that the Young function of the $\ell_{p}^{n}$ ball scales well with
the radius, that is, that $f_{i}(tx_{i})=\phi(t)f_{i}(x_{i})$ for some
function $\phi$. Of all Orlicz ball only the $\ell_{p}$ balls have this
property, which makes it impossible to apply the same proof to the generalized
Orlicz ball case.

##  4 The generalized Orlicz ball case — preliminaries, the proper measure,
lens sets

###  4.1 Idea of the proof

We would like to transfer the result given above for $\ell_{p}^{n}$ balls to
the more general case of generalized Orlicz balls. In the generalized Orlicz
ball cas the Young function does not, unfortunately, scale with the radius,
and this creates the need for a different approach. Again by Lemma 2.1 we can
restrict ourselves to characteristic functions of c-sets. As generalized
Orlicz balls are 1-symmetric, we can restrict ourselves to the positive
quadrant of our generalized Orlicz ball.

We shall proceed in two steps. The first will be to prove that generalized
Orlicz balls satisfy inequality (1.1.1) if one of the functions, say $g$, is
univariate — in other words, to begin by proving weak negative association.
This is equivalent to proving 2.1.1 for one of the sets, say $B$, being one-
dimensional. Due to Lemma 2.3, part 1, we will simply need to prove that the
function $\frac{\lambda_{n-1}(A\times\\{z\\}\cap
K)}{\lambda_{n-1}({\bar{A}}\times\\{z\\}\cap K)}$ is decreasing with $z$.
Thus, we take any $z_{2}>z_{1}\geq 0$ and concentrate on them.

We want to prove

$\frac{\lambda_{n-1}(A\times\\{z_{1}\\}\cap
K)}{\lambda_{n-1}({\bar{A}}\times\\{z_{1}\\}\cap
K)}\leq\frac{\lambda_{n-1}(A\times\\{z_{2}\\}\cap
K)}{\lambda_{n-1}({\bar{A}}\times\\{z_{2}\\}\cap K)}.$

Switching the right denominator with the left numerator we get

$\frac{\lambda_{n-1}((\\{z_{2}\\}\times A)\cap
K)}{\lambda_{n-1}((\\{z_{1}\\}\times A)\cap
K)}\geq\frac{\lambda_{n-1}((\\{z_{2}\\}\times{\bar{A}})\cap
K)}{\lambda_{n-1}((\\{z_{1}\\}\times{\bar{A}})\cap K)}$

as the inequality we need to prove. We shall denote the proportion of the
measure of $K_{z_{2}}$ to the measure of $K_{z_{1}}$ on a given set $D$ by
$\theta(D)$.

The second step will be to pass from the univariate case to the general case.
It turns out that a very similar argument, using the proportion
$\frac{\lambda(D\cap{\bar{B}})}{\lambda(D\cap B)}$ as $\theta(D)$ will allow
us to do that. Thus, to avoid repetition (as the argument is quite long), we
shall take the properties of both of these functions which make the similar
arguments possible and call any function with such properties a
$\Theta$-function, then attempt to prove

$\theta(K\cap A)\geq\theta(K)\geq\theta(K\cap{\bar{A}})$ (4.1.1)

for any $\Theta$-function $\theta$.

Section 4 is devoted to defining the concepts used in the proof (subsection
4.2) and proving general lemmas about those concepts (subsections 4.3, 4.4 and
4.5). In particular, the properties defining a $\Theta$-function are given.
Section 7 assumes inequality 4.1.1 and proves Theorem 1.2. Sections 5 and 6
are devoted to the proof of inequality 4.1.1.

The idea of Section 7 is quite simple — a Brunn-Minkowski argument and a few
approximations are enough to verify that the appropriate functions considered
for generalized Orlicz balls are in fact $\Theta$-functions. The main line of
the reasoning is similar to [W06].

To prove inequality 4.1.1 we shall attempt to divide the set $K_{+}$ into
appropriately small convex subsets $D$ for which $\theta(D)=\theta(K)$. On
each of these sets we will prove inequality (4.1.1) with $D$ substituted for
$K$, which proves the thesis ($\theta$ is a proportion, so if it is attains
some value on a family of disjoint sets, it attains the same value on the sum
of this family). The problem, of course, is to prove the inequality (4.1.1)
for any set $D$ (this is the aim of Section 5) and to construct a division
into suitable sets $D$ (this is the aim of Section 6).

For Section 5, the sets $D$ will have to be of the form
$\tilde{D}\times{\mathbb{R}}^{n-2}$, where $\tilde{D}$ is 2-dimensional.
Moreover, we will need $\tilde{D}$ to be “long and narrow”. This will allow us
to take one direction (the one in which $\tilde{D}$ is “long”) to be a new
coordinate, replacing the two coordinates of $\tilde{D}$, and to approximate
the set $A$ and the function $\theta$ on $D$ with their approximations
constant in the other, “narrow”, variable. If the approximation is good enough
(and it turns out to be), we can inductively use the inequality (4.1.1) for
the $n-1$ dimensional case for the approximating functions and then transfer
the result to the original functions.

We cannot reasonably expect the sets $D$ to have constant width in the
“narrow” coordinate. This means that in the inductive step we shall have to
consider weighted measures to take this into account. This motivates us to
consider a more general theorem, in which the Lebesgue measure on $K$ will be
replaced by a proper weighted measure.

The argument in Section 6 is somewhat similar to the Kanaan–Lovasz–Simonovits
localization lemma. However, we need the sets $D$ to satisfy additional
assumptions, in particular to be “positively inclined” (this roughly means
that the “long” coordinate axis has to be of the form $y=ax+b$, where $a$ is
positive). We were unable to fit this into the localization lemma scheme, so
the division is done by hand.

We prove in Section 5 we can cut off a “good” set $D$ from our ball.
Unfortunately, we have no control of the measure of the set we cut off (apart
from the fact it is positive). Thus inductive cutting off good sets does not
necassarily cover the whole $K$. This leads us to a transfinite inductive
reasoning, where we cut off “good” sets in a transfinite fashion (that is,
after cutting off countably many we see what is left and continue cutting).
This approach leads to a number of technical problems associated with the
limit step, and Section 6 is devoted to dealing with these problems and
following through with the transfinite induction.

###  4.2 Definitions

For the convienience of the reader all the basic definitions have been
gathered in one place. So here we will just introduce the concepts required in
the proof, and the next sections will be devoted to gaining a deeper
understanding of those concepts.

We shall usually consider a generalized Orlicz ball
$K\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$. By
$K_{x=u}$ we shall mean the section of $K_{+}$ with the hyperplane $x=u$,
similarly for any other variable in ${\mathbb{R}}^{n}$.

For a given set
$D\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$ by
${\tilde{D}}$ we shall denote the projection of $D$ to
${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$. If not said otherwise, we shall
assume $D={\tilde{D}}\times{\mathbb{R}}^{n-2}$.

######  Definition 4.1.

A function $f:{\mathbb{R}}^{n}{\rightarrow}[0,\infty)$ is called $1/m$-concave
if its support is a convex set and the function $f^{1/m}$ is concave on its
support.

######  Definition 4.2.

Let $K\subset{\mathbb{R}}^{n}$ be a generalized Orlicz ball. A measure $\mu$
on ${\mathbb{R}}^{n}$ is called a proper measure with respect to $K$ for
${\mathbb{R}}^{n}={\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$
($n\geq 2$) if the following conditions are satisfied:

  * •

$\mu$ is a non-negative measure with density $f(x)g(y){\mathbf{1}}_{K_{+}}$.

  * •

The functions $f$ and $g$ are $1/m$-concave for some $m>0$.

  * •

If $K_{x=x_{0}}=\emptyset$ for a given $x_{0}$ then $f(x_{0})=0$, and if
$K_{y=y_{0}}=\emptyset$ for a given $y_{0}$ then $g(y_{0})=0$.

In the case $n=1$ a proper measure is a non-negative measure with a
$1/m$-concave density $f$ for some $m>0$, satisfying ${\rm supp}f\subset
K_{+}$.

This definition describes the “proper weighted measures” which we will have to
analyze in the subsequent induction steps of the proof outlined above.

We shall denote the support of $f$ by $[x_{-},x_{+}]$ and the support of $g$
by $[y_{-},y_{+}]$. Of course $0\leq x_{-}\leq x_{+}$ and similarly for $y$.

If we have a proper measure on ${\mathbb{R}}^{n}$ with respect to $K$ we can
define a lens set. This definition describes the shape of a set, which will be
one of the conditions of “not losing too much on approximation” and also will
be a condition under which further dividing will be possible.

######  Definition 4.3.

A set $D\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$
is called a lens set if:

  * •

$D$ is a convex set,

  * •

$D=\tilde{D}\times{\mathbb{R}}^{n-2}$,

  * •

for some $x_{-}\leq x_{1}<x_{2}\leq x_{+}$ and $y_{-}\leq y_{1}<y_{2}\leq
y_{+}$, we have ${\tilde{D}}\subset[x_{1},x_{2}]\times[y_{1},y_{2}]$ and
$(x_{1},y_{1})\in{\tilde{D}}$ and $(x_{2},y_{2})\in{\tilde{D}}$,

  * •

$\mu(D)>0$.

A lens set is said to be a strict lens set if $x_{-}<x_{1}<x_{2}<x_{+}$,
$y_{-}<y_{1}<y_{2}<y_{+}$ and points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ are
the only points of ${\rm cl}{\tilde{D}}$ belogning to the boundary of the
rectangle $[x_{1},x_{2}]\times[y_{1},y_{2}]$.

Note that the boundary of the projection of a strict lens set onto
${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$ consists of an upper part, which is a
graph of an concave, strictly increasing function, and a lower part, which is
the graph of a convex, strictly increasing function. The boundary of a (non-
strict) lens set may additionaly contain horizontal and vertical intervals
adjacent to $(x_{1},y_{1})$ and $(x_{2},y_{2})$. We shall speak of the upper-
left border and the lower-right border of a lens set.

For a lens set $D$ we define the extremal points of ${\tilde{D}}$ to be two
points $(x_{1},y_{1})$ and $(x_{2},y_{2})$. From the definition of a lens set,
the extremal points belong to ${\tilde{D}}$. The extremal line of a lens set
is the line connecting extremal points. By the width of a lens set we shall
mean the length of its projection upon the line perpendicular to its extremal
line in the plane ${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$.

######  Definition 4.4.

For a line $L$ in ${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$ the inclination of
$L$ will denote measure of the angle between ${\mathbb{R}}_{x}$ and $L$
oriented so that the inclination of the line $\\{x=y\\}$ is $\pi/4$. A line is
said to have positive inclination if its inclination belongs to $(0,\pi/2)$,
and non-negative inclination if the inclination belongs to $[0,\pi/2]$. The
inclination of a lens set $D$ is simply the inclination of its extremal line.

By a positively inclined hyperplane in ${\mathbb{R}}^{n}$ we mean a hyperplane
$H$ defined by $x_{i}=\lambda x_{j}+c$, where $\lambda\geq 0$.

######  Definition 4.5.

For a given convex set $D$ and a proper measure $\mu$ by the relevant diameter
of $D$ we mean the diameter of $D\cap{\rm supp}\mu$.

######  Definition 4.6.

For a given generalized Orlicz ball $K\subset{\mathbb{R}}^{n}$ by its
restriction to a positively inclined hyperplane $H$ we mean such a generalized
Orlicz ball $K^{\prime}\subset{\mathbb{R}}^{n-1}$ such that $K_{+}\cap H$ is
isometric to $K_{+}^{\prime}$. By Lemma 4.19 there exists such a generalized
Orlicz ball $K^{\prime}$.

######  Definition 4.7.

For a given generalized Orlicz ball $K\subset{\mathbb{R}}^{n}$ by its
restriction to an interval $I\subset{\mathbb{R}}_{+}$ with respect to the
coordinate $x_{i}$ we mean such a generalized Orlicz ball
$K^{\prime}\subset{\mathbb{R}}^{n}$ that $K_{+}^{\prime}$ is isometric to
$K\cap\\{x_{i}\in I\\}$. By Lemma 4.18 there exists such a generalized Orlicz
ball $K^{\prime}$. When it is obvious in which coordinate the interval $I$ is
taken we shall simply write that $K^{\prime}$ it a restriction of $K$ to $I$.

######  Definition 4.8.

For a given generalized Orlicz ball $K\subset{\mathbb{R}}^{n}$ and a
generalized Orlicz ball $K^{\prime}\subset{\mathbb{R}}^{m}$ we say that
$K^{\prime}$ is a derivative of $K$ if there exists a sequence
$K=K_{0},K_{1},\ldots,K_{n}=K^{\prime}$ of generalized Orlicz balls such that
for each $i\in\\{1,\ldots,n\\}$ the ball $K_{i}$ is either a restriction of
$K_{i-1}$ to some positively inclined hyperplane or a restriction of $K_{i-1}$
with respect to some variable $x_{k}$ to some interval
$I\subset{\mathbb{R}}_{+}$.

We can embed isometrically the positive quadrant of any derivative of $K$ into
the positive quadrant of $K$. We shall identify without notice the positive
quadrant of the derivative with the image of this embedding in the positive
quadrant of $K$. In particular for a function $f$ defined on $K_{+}$ we shall
speak of its restriction to $K_{+}^{\prime}$, meaning such a function
$\tilde{f}$ that $\tilde{f}(x)=f(\phi(x))$, where $\phi$ is the embedding of
$K_{+}^{\prime}$ into $K_{+}$.

For the space ${\mathbb{R}}^{n}$ with a fixed orthonormal system
$e_{1},\ldots,e_{n}$ by a coordinate-wise decompostion of ${\mathbb{R}}^{n}$
we mean a decompostion
${\mathbb{R}}^{n}={\mathbb{R}}^{k}\times{\mathbb{R}}^{l}$, where
${\mathbb{R}}^{k}={\rm span}\\{e_{i_{1}},\ldots,e_{i_{k}}\\}$ and
${\mathbb{R}}^{l}={\rm span}\\{e_{j_{1}},\ldots,e_{j_{l}}\\}$, with $i_{p}\neq
j_{q}$ for any $p,q$.

The main tool used in this proof will be the $\Theta$ functions. We define the
$\Theta$ functions as follows:

######  Definition 4.9.

For a given generalized Orlicz ball $K\subset{\mathbb{R}}^{n}$ and two
functions $\eta_{1},\eta_{2}$ defined on $K_{+}$ we say that $\eta_{1}$ and
$\eta_{2}$ define a $\Theta$ function on $K$ if the following properties are
satisfied:

  1. T1.

The functions $\eta_{1}$ and $\eta_{2}$ are bounded.

  2. T2.

The functions $\eta_{1}$ and $\eta_{2}$ are coordinate-wise non-increasing.

  3. T3.

We have $\eta_{1}\geq\eta_{2}\geq 0$.

  4. T4.

For any derivative $K^{\prime}\subset{\mathbb{R}}^{m}$ of $K$, any proper
measure $\mu$ on $K^{\prime}$ and any coordinate-wise decomposition
${\mathbb{R}}^{m}={\mathbb{R}}^{k}\times{\mathbb{R}}^{m-k}$ the function

$\theta_{k}^{\mu}(\mathbf{x})=\frac{\int_{{\mathbb{R}}_{+}^{k}}\eta_{2}((\mathbf{y},\mathbf{x}))d\mu_{|{\mathbb{R}}^{k}}(\mathbf{y})}{\int_{{\mathbb{R}}_{+}^{k}}\eta_{1}((\mathbf{y},\mathbf{x}))d\mu_{|{\mathbb{R}}^{k}}(\mathbf{y})}$

is a coordinate-wise non-increasing function of
$\mathbf{x}=(x_{j_{1}},\ldots,x_{j_{m-k}})$ where defined. Recall
$\mu_{|{\mathbb{R}}^{k}}$ denotes the restriction of $\mu$ to
${\mathbb{R}}^{k}$.

For a fixed proper measure $\mu$ on $K$ we define the function $\theta^{\mu}$
by

$\theta^{\mu}(A)=\frac{\int_{A}\eta_{2}(x)d\mu(x)}{\int_{A}\eta_{1}(x)d\mu(x)}$

for any Borel set $A$ with $\mu(A)>0$. We shall say that $\theta^{\mu}$ is the
$\Theta$ function defined for the measure $\mu$ by $\eta_{1}$ and $\eta_{2}$.

For a fixed proper measure $\mu$ on $K$ and a fixed coordinate-wise
decomposition ${\mathbb{R}}^{n}={\mathbb{R}}^{k}\times{\mathbb{R}}^{n-k}$ we
shall also define

$\theta_{n-k}^{\mu}(a_{1},a_{2},\ldots,a_{k};A)=\frac{\int_{A}\eta_{2}(a_{1},a_{2},\ldots,a_{k},x_{k+1},x_{k+2},\ldots,x_{n})d\mu_{|\\{(a_{1},a_{2},\ldots,a_{k})\\}\times{\mathbb{R}}^{n-k}}}{\int_{A}\eta_{1}(a_{1},a_{2},\ldots,a_{k},x_{k+1},x_{k+2},\ldots,x_{n})d\mu_{|\\{(a_{1},a_{2},\ldots,a_{k})\\}\times{\mathbb{R}}^{n-k}}}$

for such sets $A$ and number $a_{1},a_{2},\ldots,a_{k}$ for which the
denominator is positive. If $A={\mathbb{R}}^{n-k}$ we shall omit it and write
$\theta_{n-k}^{\mu}(a_{1},a_{2},\ldots,a_{k})$ for
$\theta_{n-k}^{\mu}(a_{1},a_{2},\ldots,a_{k};{\mathbb{R}}^{n-k})$, and if
$\mathbf{a}=(a_{1},a_{2},\ldots,a_{k})$, we will write
$\theta_{n-k}^{\mu}(\mathbf{a};A)$ or $\theta_{n-k}^{\mu}(\mathbf{a})$ for
$\theta_{n-k}^{\mu}(a_{1},a_{2},\ldots,a_{k};A)$ and
$\theta_{n-k}^{\mu}(a_{1},a_{2},\ldots,a_{k};{\mathbb{R}}^{n-k})$
respectively, which is consistent with the notation above. If
$A\subset{\mathbb{R}}^{n}$ by $\theta^{\mu}_{n-k}(\mathbf{a};A)$ we mean
$\theta^{\mu}_{n-k}(\mathbf{a};A\cap\\{\mathbf{a}\\}\times{\mathbb{R}}^{n-k})$.
If there could be doubts as to what coordinate-wise decomposition is taken, we
may write $\theta_{n-k}^{\mu}(x_{1}=a_{1},x_{1}=a_{2},\ldots,x_{k}=a_{k};A)$
for $\theta_{n-k}^{\mu}(a_{1},a_{2},\ldots,a_{k};A)$.

######  Fact 4.10.

If $\eta_{1}$ and $\eta_{2}$ define a $\Theta$ function $\theta^{\mu}$ for a
proper measure $\mu$ on a generalized Orlicz ball $K$, then the following are
true:

  1. T5.

The function $\theta^{\mu}$ is continuous with respect to the symmetric
difference distance, that is if $\theta^{\mu}$ is defined for all $C_{i}$ and
$\mu(C_{0}\bigtriangleup C_{i}){\rightarrow}0$, then
$\theta^{\mu}(C_{i}){\rightarrow}\theta^{\mu}(C_{0})$.

  2. T6.

If $D^{\prime}\subset D\subset{\mathbb{R}}^{n}$,
$\theta^{\mu}(D)=\theta^{\mu}(D^{\prime})$ and $\theta^{\mu}$ is defined for
$D\setminus D^{\prime}$, then $\theta^{\mu}(D\setminus
D^{\prime})=\theta^{\mu}(D)$.

  3. T7.

If $K^{\prime}$ is a derivative of $K$, then the restrictions of $\eta_{1}$
and $\eta_{2}$ to $K^{\prime}$ define a $\Theta$ function on $K^{\prime}$.

Further on, as the proper measure taken rarely changes, we omit the $\mu$ in
the upper index and simply write $\theta$ for $\theta^{\mu}$. Note that as
$\eta_{1}$ is positive on $K_{+}$ from property (T3) and ${\rm supp}\mu\subset
K_{+}$, we know that $\theta^{\mu}(D)$ is well defined if and only if
$\mu(D)>0$.

######  Definition 4.11.

Functions $\eta_{1}$ and $\eta_{2}$ defining a $\Theta$ function on a
generalized Orlicz ball $K$ are said to define a strict $\Theta$ function if
the following extra conditions are satisfied:

  1. S1.

${\rm supp}\eta_{2}\subset{\rm Int}_{{\mathbb{R}}_{+}}{\rm supp}\eta_{1}$,
where ${\rm Int}_{{\mathbb{R}}_{+}}$ denotes the interior taken with respect
to the space ${\mathbb{R}}_{+}$.

  2. S2.

The generalized Orlicz ball $K$ is proper.

  3. S3.

For any coordinate-wise decomposition
${\mathbb{R}}^{n}={\mathbb{R}}^{k}\times{\mathbb{R}}^{n-k}$ with $n>k$ the
functions ${\tilde{\eta}}_{i}:{\mathbb{R}}^{k}{\rightarrow}{\mathbb{R}}$
defined by $x\mapsto\int_{{\mathbb{R}}^{n-k}}\eta_{i}(x,y)d\lambda_{n-k}(y)$
are continuous.

  4. S4.

$\eta_{1}>0$ on ${\rm Int}K_{+}$.

######  Definition 4.12.

For $\eta_{1}$ and $\eta_{2}$ defining a $\Theta$ function $\theta$ on a
generalized Orlicz ball $K$ by a derivative of $\theta$ we mean the function
defined on a derivative $K^{\prime}$ of $K$ by the restrictions of $\eta_{1}$
and $\eta_{2}$ to $K^{\prime}$. Note that the derivatives of a $\Theta$
function are $\Theta$ functions.

######  Definition 4.13.

For a given generalized Orlicz ball $K$ we say that $\eta_{1}$ and $\eta_{2}$
define a weakly non-degenerate $\Theta$ function on $K$ if for every
${\varepsilon}>0$ there exists a generalized Orlicz ball $K^{\prime}\subset K$
with $\lambda(K\setminus K^{\prime})\leq{\varepsilon}\lambda(K)$ and functions
$\eta_{1}^{\prime}$ and $\eta_{2}^{\prime}$ defining a strict $\Theta$
function on $K^{\prime}$ with
$\int|\eta_{i}-\eta_{i}^{\prime}|d\lambda\leq{\varepsilon}$. A $\Theta$
function is called non-degenerate if it is weakly non-degenerate and all its
derivatives are weakly non-degenerate.

Note that as the density of any proper measure is bounded, in all the bounds
in the definition above we can replace $\lambda$ by any proper measure $\mu$.

Frequently we shall take the same collection of assumptions for our theorems.
To make reading the paper easier, we will use the following notation:

######  Definition 4.14.

We shall speak of

  * •

Standard assumptions if
$K\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$ is a
generalized Orlicz ball, $\mu$ is a proper measure for $K$, $\eta_{1}$ and
$\eta_{2}$ define a $\Theta$ function $\theta=\theta^{\mu}$ on $K$ for $\mu$
and $A$ is a c-set in ${\mathbb{R}}_{+}^{n}$,

  * •

Non-degenerate assumptions if additionally we require the $\Theta$ function
defined by $\eta_{1}$ and $\eta_{2}$ to be non-degenerate, and

  * •

Strict assumptions if $K$ is a proper generalized Orlicz ball and $\eta_{1}$
and $\eta_{2}$ define a strict non-degenerate $\Theta$ function.

######  Definition 4.15.

Under standard assumptions a set $D\subset{\mathbb{R}}^{n}$ will be called
appropriate, if

  * •

$\theta(D)$ is defined,

  * •

$\theta(D\cap A)\geq\theta(K)\geq\theta(D\cap{\bar{A}})$ if the left-hand side
and the right-hand side are defined,

  * •

$\theta(D)=\theta(K)$.

######  Definition 4.16.

Under standard assumptions let $\mu_{2}$ be the restriction of $\mu$ to
${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times\\{0\\}$. For any
${\varepsilon}>0$ a set ${\tilde{D}}\times{\mathbb{R}}^{n-2}=D$ is called
${\varepsilon}$-appropriate, if

  * •

$\theta(D)$ is defined,

  * •

$\theta(D)=\theta(K)$,

  * •

For each $U\in\\{A,{\bar{A}}\\}$ and each $i\in\\{1,2\\}$ there exists a
number $C_{U,i}$ such that

$\bigg{|}\int_{D\cap
U}\eta_{i}(t)d\mu(t)-C_{U,i}\bigg{|}\leq{\varepsilon}\mu_{2}({\tilde{D}})$

and

$\frac{C_{A,2}}{C_{A,1}}\geq\theta(D)\geq\frac{C_{{\bar{A}},2}}{C_{{\bar{A}},1}}.$

The definition of an appropriate set describes the properties we desire for
the set into which we divide $K_{+}$. In fact, due to the approximation, we
shall divide $K_{+}$ into ${\varepsilon}$-appropriate sets to prove it is
${\varepsilon}$-appropriate, and then take ${\varepsilon}{\rightarrow}0$.

###  4.3 The generalized Orlicz ball lemmas

In this subsection we will prove a few lemmas about the structure generalized
Orlicz balls. They show that the class of generalized Orlicz balls is closed
under taking derivatives, and that proper generalized Orlicz balls are, in a
sense, dense in the class of generalized Orlicz balls. These lemmas are the
main reason the whole reasoning in this paper has to be done for generalized
Orlicz balls, and not simply Orlicz balls — the class of Orlicz balls does not
enjoy the same closedness propeties.

######  Fact 4.17.

A product of intervals $\prod_{i=1}^{n}[a_{i},b_{i}]$ is isometric to the
positive quadrant of the Orlicz ball $K\subset{\mathbb{R}}^{k}$ defined by the
functions

$\displaystyle f_{i}(x_{i})=\begin{cases}0&\hbox{ if $x_{i}\leq
b_{i}-a_{i}$}\\\ \infty&\hbox{ if $x_{i}>b_{i}-a_{i}$}\end{cases}$

for $b_{i}>a_{i}$.

######  Lemma 4.18.

If $K_{+}\subset{\mathbb{R}}^{n}$ is a generalized Orlicz ball positive
quadrant and $0\leq x_{a}<x_{b}$, then $K_{+}\cap\\{x_{1}\in[x_{a},x_{b}]\\}$
is isometric to a generalized Orlicz ball positive quadrant or empty

###### Proof.

Let $f_{1},f_{2},\ldots,f_{n}$ be the Young functions defining $K$. Let
$K_{+}^{\prime}=K_{+}\cap\\{x_{1}\in[x_{a},x_{b}]\\}$. Let $c=f_{1}(x_{a})$.
If $c=1$, then
$K_{+}^{\prime}=\\{x:x_{1}=x_{a},\forall_{i>1}f_{i}(x_{i})=0\\}$, which is a
product of intervals and thus isometric to a generalized Orlicz ball positive
quadrant. If $c>1$ then $K_{+}^{\prime}$ is empty. If $c<1$ we define
$\bar{f}_{1}$ by

$\displaystyle\bar{f}_{1}(x_{1})=\begin{cases}\frac{f_{1}(x_{1}+x_{a})-f_{1}(x_{a})}{1-c}&\hbox{
for $x_{1}<x_{b}$}\\\ \infty&\hbox{ for $x_{1}>x_{b}$,}\end{cases}$

and $\bar{f}_{i}$ for $i>1$ by

$\bar{f}_{i}(x_{i})=\frac{f_{i}(x_{i})}{1-c}.$

Now $(x_{1},x_{2},\ldots,x_{n})\in\bar{K}_{+}$ iff
$(x_{1}+x_{a},x_{2},\ldots,x_{n})\in K_{+}^{\prime}$, where $\bar{K}_{+}$ is
the positive quadrant of the Orlicz ball defined by $\bar{f}_{i}$. ∎

######  Lemma 4.19.

If $K\subset{\mathbb{R}}^{n}$ is a generalized Orlicz ball and
$H=\\{x\in{\mathbb{R}}^{n}:x_{1}=\lambda x_{2}+c\\}$ is a positively inclined
hyperplane in ${\mathbb{R}}^{n}$, then $K_{+}\cap H$ is the positive quadrant
of some generalized Orlicz ball $L$ or an empty set.

###### Proof.

As $H$ is positively inclined, $\lambda\geq 0$. If $\lambda=0$ and $c<0$ we
have $K_{+}\cap H=\emptyset$. If $c<0$ and $\lambda>0$ we can transform the
equation giving $H$ to
$H=\\{x\in{\mathbb{R}}^{n}:x_{2}=\frac{1}{\lambda}x_{1}-\frac{c}{\lambda}\\}$.
Thus we can assume $c\geq 0$.

For $x_{2}\geq 0$ we have $x_{1}\geq c$ in $H$. Thus if $f_{1}(c)>1$, then for
$x_{2}\geq 0$ we have $f_{1}(x_{1})>1$ for $x\in H,x_{1}\geq 0$, thus $H\cap
K_{+}=\emptyset$. If $f_{1}(c)=1$ and $\lambda>0$, then $H\cap K_{+}$ is the
set $\\{x:x_{1}=c,x_{2}=0,f_{i}(x_{i})=0$ for $i>2\\}$. This set is a
cartesian product of intervals, and isometric to a generalized Orlicz ball
positive quadrant. If $f_{1}(c)=1$ and $\lambda=0$, the situation is the same,
except $x_{2}=0$ is replaced by $f_{2}(x_{2})=0$.

Now we may assume $f_{1}(c)<1$. Let $x_{3},x_{4},\ldots,x_{n+1}$ be the
coordinates on $H$, with $x_{1}=\lambda x_{n+1}+c$, $x_{2}=x_{n+1}$. Let us
take $f_{n+1}(t)=f_{1}(\lambda t+c)+f_{2}(t)-f_{1}(c)$, then
$f_{1}(x_{1})+f_{2}(x_{2})=f_{n+1}(x_{2})+f_{1}(c)$. The function $f_{n+1}$ is
a sum of three convex functions, thus it is convex, and $f_{n+1}(0)=0$. The
set $\\{(x_{i})_{i=3}^{n+1}:f_{i}(x_{i})<1-f_{1}(c)\\}\cap{\mathbb{R}}_{+}$ is
equal to $K_{+}\cap H$. If we consider Young functions
$\tilde{f}_{i}(t)=\frac{f_{i}(t)}{1-f_{1}(c)}$ for $i=3,4,\ldots,n+1$ we get
the generalized Orlicz ball $L\subset H$ such that $K_{+}\cap
H=L\cap{\mathbb{R}}_{+}^{n}$.∎

######  Lemma 4.20.

For any generalized Orlicz ball $K\subset{\mathbb{R}}^{n}$ and any
${\varepsilon}>0$ there exists a proper generalized Orlicz ball
$K^{\prime}\subset K$ with $\lambda(K\setminus K^{\prime})<{\varepsilon}$.
Furthermore if any Young function $f_{i}$ of $K$ is already a proper Young
function, the same $f_{i}$ will be the appropriate Young function of
$K^{\prime}$.

###### Proof.

This lemma is easy to believe in, but somewhat technical to prove. An
impatient reader might be well advised to skip the next two proofs (or prove
the Lemmas her$/$himself, if desired) and go to the more crucial parts of the
paper.

As any generalized Orlicz ball is 1-symmetric, it suffices to prove
$\lambda(K_{+}\setminus K^{\prime}_{\geq 0})\leq{\varepsilon}/2^{n}$. We shall
thus consider only the points in ${\mathbb{R}}_{+}^{n}$ and decrease
${\varepsilon}$ to be $2^{n}$ times smaller. Recall that a proper Young
function is such a Young function that $f(x)=0$ only for $x=0$ and
$f(x)<\infty$. Thus we have to get rid of superfluous zeroes and of infinity
values. First we shall take care of the zeroes.

Let $f_{i}$ be Young functions defining $K$. Let $M$ be the largest of the
$(n-1)$-dimensional measures of the projections of $K$ onto the hyperplanes
$x_{i}=0$. Let $t_{i}=\inf\\{x_{i}:f_{i}(x_{i})>1/2n\\}$. Let
$c=\inf_{i}\\{f_{i}^{\prime}(t_{i})\\}$. We shall prove that for $\delta<1/2n$
the set $U_{\delta}=\\{x:\sum f_{i}(x_{i})\in[1-\delta,1]\\}$ has measure no
larger than $\frac{Mn\delta}{c}$.

First note that $U_{\delta}=\bigcup U_{i}$, where
$U_{i}=U_{\delta}\cap\\{x:f_{i}(x_{i})>1/2n\\}$, as at least one of
$f_{i}(x_{i})$ has to be large for the sum to be large. We shall bound the
measure of each $U_{i}$ separately. For each point
$x=(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{n})$ the set of those $x_{i}$ that
$(x,x_{i})\in U_{i}$ has length at most $\frac{\delta}{c}$. Thus, from
Fubini’s theorem, the measure of $U_{i}$ can be bounded by
$\frac{M\delta}{c}$, and summing over all $i$ we get the desired bound for
$U_{\delta}$.

Let us take $\delta=\frac{c{\varepsilon}}{2Mn^{2}}$. For each $i$ for which we
have superfluous zeroes let us take
$s_{i}=\inf\\{x_{i}:f_{i}(x_{i})>\delta\\}$ and replace $f_{i}$ by $g_{i}$
defined by

$g_{i}(x_{i})=\begin{cases}f_{i}(x_{i})&\hbox{ for }x_{i}\geq s_{i}\\\
x_{i}\delta/s_{i}&\hbox{ for }x_{i}<s_{i}.\end{cases}$

We have $g_{i}(x_{i})\geq f_{i}(x_{i})$ and
$g_{i}(x_{i})-f_{i}(x_{i})\leq\delta$. Thus if $K^{\prime}$ is the generalized
Orlicz ball defined by $g_{i}$, we have $K^{\prime}\subset K$ and $K\setminus
K^{\prime}\subset U_{n\delta}$, and thus $\lambda(K\bigtriangleup
K^{\prime})\leq\frac{n^{2}M\delta}{c}={\varepsilon}/2$.

Now we shall deal with the $\infty$ values. Let
$\delta=\frac{{\varepsilon}}{2nM}$. Note that the shape of $K^{\prime}$ is
determined by the values of $g_{i}$ only up to $g_{i}(x_{i})=1$. Thus we have
to make some corrections to $g_{i}$ up to $g_{i}(x_{i})=1$, and then extend
$g_{i}$ anyhow, say linearly. For each $i$ such that $g_{i}$ attains the
$\infty$ value let $r_{i}=\inf\\{x_{i}:g_{i}(x_{i})=\infty\\}$, and let
$v_{i}=\lim_{x_{i}{\rightarrow}v_{i}^{-}}g_{i}(x_{i})$. If $v_{i}\geq 1$, then
all we have to do is to extend $g_{i}$ in a different way after $r_{i}$, and
that does not change the ball $K^{\prime}$ defined by $g_{i}$. If, however,
$v_{i}<1$, we define $h_{i}$ as follows:

$h_{i}(x_{i})=\begin{cases}g_{i}(x_{i})&\hbox{ for }x_{i}<r_{i}-\delta\\\
2&\hbox{ for }x_{i}=r_{i}\\\ \hbox{linear continuous extension}&\hbox{
otherwise.}\end{cases}$

Let $K^{\prime\prime}$ be the ball defined by $h_{i}$. Again,
$K^{\prime\prime}\subset K^{\prime}$, as $h_{i}\geq g_{i}$ on the set where
$g_{i}\geq 1$, from the convexity of $g_{i}$. The difference, however, is
obviously contained in
$\bigcup_{i}K^{\prime}\cap\\{x_{i}\in[r_{i}-\delta,r_{i}]\\}$, thus
$\lambda(K^{\prime}\setminus K^{\prime\prime})\leq nM\delta={\varepsilon}/2$.
Adding the two estimates together we get $\lambda(K\setminus
K^{\prime\prime})\leq{\varepsilon}$. ∎

######  Corollary 4.21.

With the assumptions of Lemma 4.20 if we take any $y_{0}$ (where $y$ is any
coordinate in ${\mathbb{R}}^{n}$), then we can take such a $K^{\prime}$ as
before and $y_{1}$ that
$\lambda_{n-1}((K\cap\\{y=y_{0}\\})\bigtriangleup(K^{\prime}\cap\\{y=y_{1}\\}))<{\varepsilon}$.

###### Proof.

This, again, is easy to believe in, and actually simple if $f_{y}(y_{0})\neq
1$. The special case where $f_{y}(y_{0})=1$ could arguably be ignored (as it
happens only on a set of measure zero), but to avoid omitting a set of measure
zero in all other places of the proof, we shall go through the technicalities
here.

If $f_{y}(y_{0})>1$, we can simply take $y_{1}=y_{0}$, and
$\lambda_{n-1}(K\cap\\{y=y_{0}\\})=\lambda_{n-1}(K^{\prime}\cap\\{y=y_{1}\\})=0$.

If $f_{y}(y_{0})<1$, we need to control the Orlicz ball $\sum
f_{i}(x_{i})=1-f_{y}(y_{0})$. This Orlicz ball $L$ is given by Young functions
$f_{i}/(1-f_{y}(y_{0}))$. For this Orlicz ball we also calculate values of $M$
and $c$, and apply the reasoning in the proof of Lemma 4.20 taking the larger
$M$ and the smaller $c$ of those calculated for the two balls. We thus get
good approximations $K^{\prime}$ and $L^{\prime}$ of both $K$ and $L$. Now
take such a $y_{1}$ that $f_{y}(y_{0})=h_{y}(y_{1})$, this can be done as
$h_{y}$ is continuous. Now $K^{\prime}\cap\\{y=y_{1}\\}=L^{\prime}$, which
proves the thesis.

In the case $f_{y}(y_{0})=1$ if any of the other $f_{i}$ do not have
superfluous zeroes, the measure of $K\cap\\{y=y_{0}\\}$ is 0, and thus taking
$y_{1}=y_{0}+1$ we get the thesis. If, however, all the other $f_{i}$ have
superfluous zeroes, the intersection $K\cap\\{y=y_{0}\\}$ is the cube
${\prod}_{i}f_{i}^{-1}(0)$. In this case we shall need a better approximation.
Let $z_{i}=\sup\\{x_{i}:f_{i}(x_{i})=0\\}$. Let us, as before for
${\varepsilon}$, define
$\delta^{\prime}=\frac{c{\varepsilon}^{\prime}}{2Mn^{2}}$ and
$s_{i}^{\prime}=\inf\\{x_{i}:f_{i}(x_{i})>\delta^{\prime}\\}$. We need
${\varepsilon}^{\prime}$ to be so small that

$s_{i}^{\prime}/z_{i}\leq 1+\frac{(1+{\varepsilon}/M)^{1/n}-1}{n}$

and smaller than ${\varepsilon}$. Note that as
${\varepsilon}^{\prime}{\rightarrow}0$ we have $\delta^{\prime}{\rightarrow}0$
and $s_{i}^{\prime}{\rightarrow}z_{i}$, so taking ${\varepsilon}^{\prime}$
small enough we can achieve the desired inequality for all $i$. Conduct the
proof of Lemma 4.20 taking ${\varepsilon}^{\prime}$ instead of
${\varepsilon}$. Take $y_{1}$ such that $h_{y}(y_{1})=1-n\delta^{\prime}$.
Note that if $x_{i}\leq s_{i}^{\prime}$ for all $i$, then $\sum
h_{i}(x_{i})\leq n\delta^{\prime}$, and thus $x=(x_{i})\in
K^{\prime}\cap\\{y=y_{1}\\}$. On the other hand if for any $i$ we have
$x_{i}>s_{i}^{\prime}+(n-1)(s_{i}^{\prime}-z_{i})$, then $h_{i}(x_{i})\geq
f_{i}(x_{i})$, and as $f_{i}(z_{i})=0$,
$f_{i}(s_{i}^{\prime})\geq\delta^{\prime}$ and $f_{i}$ is convex, we have
$f_{i}(x_{i})>n\delta^{\prime}$, and thus $\sum h_{i}(x_{i})>n\delta^{\prime}$
and $x\not\in K^{\prime}\cap\\{y=y_{1}\\}$. Thus

$K\cap\\{y=y_{0}\\}={\prod}_{i}f_{i}^{-1}(0)\subset
K^{\prime}\cap\\{y=y_{1}\\}\subset{\prod}_{i}[0,z_{i}+n(s_{i}^{\prime}-z_{i})].$

Now we have the following inequalities:

$\displaystyle\frac{s_{i}^{\prime}}{z_{i}}$ $\displaystyle\leq
1+\frac{(1+{\varepsilon}/M)^{1/n}-1}{n}$
$\displaystyle\frac{n(s_{i}^{\prime}-z_{i})}{z_{i}}$
$\displaystyle\leq(1+{\varepsilon}/M)^{1/n}-1$
$\displaystyle\frac{z_{i}+n(s_{i}^{\prime}-z_{i})}{z_{i}}$
$\displaystyle\leq(1+{\varepsilon}/M)^{1/n}$
$\displaystyle\prod_{i}\frac{z_{i}+n(s_{i}^{\prime}-z_{i})}{z_{i}}$
$\displaystyle\leq 1+{\varepsilon}/M$
$\displaystyle\lambda({\prod}_{i}[0,z_{i}+n(s_{i}^{\prime}-z_{i})])$
$\displaystyle\leq\lambda({\prod}[0,z_{i}])+\frac{{\varepsilon}\lambda({\prod}[0,z_{i}])}{M}$
$\displaystyle\lambda(K^{\prime}\cap\\{y=y_{1}\\})-\lambda(K\cap\\{y=y_{0}\\})$
$\displaystyle\leq{\varepsilon}.$

The last inequality follows as ${\prod}[0,z_{i}]$ is a subset of the
projection of $K$ onto $y=0$, and thus its measure is no bigger than $M$.
This, along with the fact that $K\cap\\{y=y_{0}\\}\subset
K^{\prime}\cap\\{y=y_{1}\\}$ gives the thesis.

This reasoning can be extended to approximate any finite number of sections of
$K$ along with $K$. ∎

###  4.4 $1/m$-concave functions and proper measures

Here we give a few elementary facts about $1/m$-concave functions and proper
measures. Most facts are easily proved and quite a few are well known, so we
skip some of the proofs.

######  Fact 4.22.

If a function $f$ is $1/m$-concave for some $m>0$, then it is also
$1/m^{\prime}$-concave for any $m^{\prime}>m$.

######  Fact 4.23.

The product of $1/m$-concave functions is $1/2m$-concave.

######  Fact 4.24.

From the Brunn-Minkowski inequality, if $K\subset{\mathbb{R}}^{n}$ is a convex
set, then
$(y_{1},y_{2},\ldots,y_{k})\mapsto\lambda_{n-k}(K\cap\\{\forall_{1\leq i\leq
k}x_{i}=y_{i}\\})$ is a $1/(n-k)$-concave function, where $x_{i}$ are the
coordinates on ${\mathbb{R}}^{n}$. Conversely, if we have a $1/m$-concave
function on ${\mathbb{R}}^{n}$, then there exists a convex set
$K\subset{\mathbb{R}}^{n+m}$ such that $f$ is the projection of the Lebesgue
measure restricted to $K$ onto ${\mathbb{R}}^{n}$. As a corollary of these two
facts the projection of a $1/m$ concave function on ${\mathbb{R}}^{n}$ onto
${\mathbb{R}}^{k}$ is a $1/(n+m-k)$-concave function.

######  Fact 4.25.

The restriction of the Lebesgue measure to $K_{+}$ is a proper measure with
respect to $K$.

######  Fact 4.26.

The support of a proper measure $\mu$ is a convex set.

######  Lemma 4.27.

If $\mu$ is a proper measure on ${\mathbb{R}}^{n}$ and
$H=\\{x\in{\mathbb{R}}^{n}:x_{1}=\lambda x_{2}+c\\}$, with $\lambda\geq 0$ is
a hyperplane in ${\mathbb{R}}^{n}$, then $\mu$ restricted to $H$ with
coordinates $(v,x_{3},\ldots,x_{n})$ is a proper measure.

###### Proof.

The density of $\mu$ is equal to $f(x)g(y){\mathbf{1}}_{K}$ for some
$(1/m)$-concave functions $f$ and $g$ and some generalized Orlicz ball $K$.
From Lemma 4.19 the set $K_{+}\cap H$ is the positive quadrant of some Orlicz
ball $K^{\prime}$ with coordinates $(v,x_{3},\ldots,x_{n})$. The product
$f\cdot g$ on $H$ varies with at most two variables, and is from, Fact 4.23, a
$(1/2n)$-concave function with respect to $v$. ∎

######  Lemma 4.28.

If $\mu$ is a proper measure on ${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times
R^{n-2}$, then the restriction of $\mu$ to an interval $I$ with respect to any
variable is also a proper measure.

###### Proof.

Due to Lemma 4.18 if $K_{+}$ is the Orlicz ball quadrant for which $\mu$ is
defined, $K_{+}^{\prime}=K_{+}\cap\\{t\in I\\}$ is also an Orlicz ball
quadrant. Let $f$ and $g$ be the functions defining the density of $\mu$. To
make them define a proper measure on $K_{+}^{\prime}$ we simply have to
restrict them to the set $\\{x_{0}:\lambda_{n-1}(K^{\prime}_{x=x_{0}})>0\\}$
for $f$ and similarly for $g$, and additionaly to the interval $I$ if it was
taken in $x$ or $y$ respectively. Both functions will have a convex support
after this restriction, and as they were $1/m$-concave on a larger domain,
they will still be $1/m$-concave. ∎

###  4.5 Lens sets and $\Theta$ functions

######  Fact 4.29.

Let $D$ be a lens set with extremal points $(x_{1},y_{1})$ and
$(x_{2},y_{2})$. Let $x^{-}(y)=\inf\\{x:(x,y)\in D\\}$,
$x^{+}(y)=\sup\\{x:(x,y)\in D\\}$ for $y\in[y_{1},y_{2}]$. Then $x^{-}$ and
$x^{+}$ are increasing function on their domains, $x^{-}$ is convex, and
$x^{+}$ is concave.

######  Lemma 4.30.

Under standard assumptions consider a fixed $y_{0}$ and two intervals
$[x_{a},x_{b}],[x_{c},x_{d}]$ with $x_{a}\leq x_{c}$ and $x_{b}\leq x_{d}$.
Then we have

$\theta^{\mu}_{n-1}(y_{0};[x_{a},x_{b}]\times{\mathbb{R}}^{n-2})\geq\theta^{\mu}_{n-1}(y_{0};[x_{c},x_{d}]\times{\mathbb{R}}^{n-2})$

if both sides are defined.

The same applies when $x$ is exchanged with $y$.

###### Proof.

From property (T4) we know that $\theta^{\mu}_{n-2}(x,y_{0})$ is a decreasing
function of $x$. The domain of this function is a convex set, so its
intersections with $[x_{a},x_{b}]$ and $[x_{c},x_{d}]$ are both intervals
(they are non-empty, for $\theta^{\mu}_{n-1}$ is defined for both intervals).
Applying Lemma 2.3, part 2, to
$\int_{{\mathbb{R}}^{n-2}}\eta_{2}(x,y,t_{1},\ldots,t_{n-2})d\mu_{|(x,y)\times{\mathbb{R}}^{n-2}}(t_{1},\ldots,t_{n-2})$
and
$\int_{{\mathbb{R}}^{n-2}}\eta_{1}(x,y,t_{1},\ldots,t_{n-2})d\mu_{|(x,y)\times{\mathbb{R}}^{n-2}}(t_{1},\ldots,t_{n-2})$
and the shortened intervals we get the thesis. ∎

######  Lemma 4.31.

Under standard assumptions for a given interval $I=[x_{a},x_{b}]$ the function
$\theta^{\mu}_{n-1}(y;I\times{\mathbb{R}}^{n-2})$ is a decreasing function of
$y$ on its domain. The same applies when $x$ is exchanged with $y$.

###### Proof.

Take any $0\leq y_{1}\leq y_{2}$ in the domain. $K^{\prime}=K\cap\\{x\in I\\}$
is a derivative of $K$, and
$\theta^{\mu}_{n-1}(y;I\times{\mathbb{R}}^{n-2})=\bar{\theta}_{n-1}^{\mu}(y)$,
where $\bar{\theta}^{\mu}$ is defined by the restrictions of $\eta_{1}$ and
$\eta_{2}$ to $K^{\prime}$. Thus from property (T4) we get the thesis. ∎

######  Lemma 4.32.

Under standard assumptions for a given lens set $D$ the domain of the function
$y\mapsto\theta^{\mu}_{n-1}(y;D)$ is an interval and the function is
decreasing.

###### Proof.

Let $(x_{1},y_{1})$ and $(x_{2},y_{2})$ be the extremal points of
${\tilde{D}}$. Take any $y_{4}$ such that $\theta_{n-1}^{\mu}(y_{4};D)$ is
defined and take any $y_{3}\in(y_{1},y_{4}]$. Thus
$\theta_{n-2}^{\mu}(x,y_{4})$ is defined for more than one $x$ such that
$(x,y_{4})\in{\tilde{D}}$ (actually, for a set of positive Lebesgue measure),
let $x_{4}$ be any such $x$ except the smallest. We want to prove
$\theta_{n-1}^{\mu}(y_{3};D)$ is defined. Note that ${\rm supp}\mu$ is a c-set
on $x>x_{-}$, $y>y_{-}$ and ${\rm supp}\eta_{1}$ is also a c-set, thus their
intersection is a c-set. Thus $\theta_{n-2}^{\mu}(x,y)$ is defined for any
$x_{-}<x\leq x_{4}$ and $y_{-}<y\leq y_{4}$. As $D$ is a lens set, the set of
$x\leq x_{4}$ such that $(x,y_{3})\in{\tilde{D}}$ has positive Lebesgue
measure, thus $\theta_{n-1}^{\mu}(y_{3};D)$ is defined, which means that
$\theta_{n-1}^{\mu}(y;D)$ is defined on some interval $(y_{1},y_{0})$ and
undefined outside.

Now we shall prove $\theta_{n-1}^{\mu}(y;D)$ is decreasing. Take $y_{3}\leq
y_{4}$ from the domain. Let $[x_{3}^{-},x_{3}^{+}]$ be the interval
${\tilde{D}}\cap\\{y=y_{3}\\}$ and $[x_{4}^{-},x_{4}^{+}]$ the interval
${\tilde{D}}\cap\\{y=y_{4}\\}$. From the definition of a lens set
$x_{3}^{-}\leq x_{4}^{-},x_{3}^{+}\leq x_{4}^{+}$. From Lemmas 4.31 and 4.30
(twice) we have

$\displaystyle\theta_{n-1}^{\mu}(y_{3};D)$
$\displaystyle=\theta_{n-1}^{\mu}(y_{3};[x_{3}^{-},x_{3}^{+}]\times{\mathbb{R}}^{n-2})\geq\theta_{n-1}^{\mu}(y_{3};[x_{3}^{-},x_{4}^{+}]\times{\mathbb{R}}^{n-2})$
$\displaystyle\geq\theta_{n-1}^{\mu}(y_{4};[x_{3}^{-},x_{4}^{+}]\times{\mathbb{R}}^{n-2})\geq\theta_{n-1}^{\mu}(y_{4};[x_{4}^{-},x_{4}^{+}]\times{\mathbb{R}}^{n-2})=\theta_{n-1}^{\mu}(y_{4};D).$

Note that the last expression in the first line and the first in the second
line are well defined, for the second argument is a superset of the second
argument for $\theta_{n-1}^{\mu}(y_{3};D)$ and $\theta_{n-1}^{\mu}(y_{4};D)$
respectively. ∎

######  Corollary 4.33.

Under standard assumptions for a given lens set $D$ and a given $y_{0}$ in the
domain of $\theta_{n-1}^{\mu}(y;D)$ we have

$\theta^{\mu}(D\cap\\{y\leq
y_{0}\\})\leq\theta_{n-1}^{\mu}(y_{0};D)\leq\theta^{\mu}(D\cap\\{y\geq
y_{0}\\})$

and

$\theta^{\mu}(D\cap\\{y\leq
y_{0}\\})\leq\theta^{\mu}(D)\leq\theta^{\mu}(D\cap\\{y\geq y_{0}\\})$

if the left and right hand sides are defined.

Moreover, if $\theta^{\mu}(D\cap\\{y\leq y_{0}\\})=\theta^{\mu}(D\cap\\{y\geq
y_{0}\\})$ for any $y_{0}\in(y_{1},y_{2})$, then
$\theta^{\mu}_{n-1}(y_{3};D),\theta^{\mu}(D\cap\\{y\geq y_{3}\\})$ and
$\theta^{\mu}(D\cap\\{y\leq y_{3}\\})$ are all constant where defined and
equal $\theta^{\mu}(D)$ for $y_{3}\in(y_{1},y_{2})$.

###### Proof.

From Lemma 4.32 the function $\theta_{n-1}^{\mu}(y;D)$ is decreasing as a
function $y$ on its domain, and its domain is an interval. We know that ${\rm
supp}\eta_{2}\subset{\rm supp}\eta_{1}$, so we can apply Lemma 2.3, part 1a,
to the appropriate integrals of $\eta_{2}$ and $\eta_{1}$ to get the first
part of the thesis. The second part follows from the first and Fact 2.2. The
third follows again from Lemma 2.3, part 1b. ∎

######  Proposition 4.34.

Under standard assumptions if $D$ is ${\varepsilon}$-appropriate for any
${\varepsilon}>0$, then $D$ is appropriate.

###### Proof.

The third and first condition in Definition 4.15 follows from the definition
of ${\varepsilon}$-appropriate for any ${\varepsilon}$. We have to check the
second condition. Let $C_{U,i}^{\varepsilon}$ denote the numbers $C_{U,i}$
which show $D$ is and ${\varepsilon}$-appropriate set. We have

$\theta^{\mu}(D)=\frac{C_{A,2}^{\varepsilon}}{C_{A,1}^{\varepsilon}}\leq\frac{\int_{D\cap
A}\eta_{2}(t)d\mu(t)+{\varepsilon}\mu_{2}({\tilde{D}})}{\int_{D\cap
A}\eta_{1}(t)d\mu(t)-{\varepsilon}\mu_{2}({\tilde{D}})}\rightarrow_{{\varepsilon}{\rightarrow}0}\theta^{\mu}(D\cap
A),$

and similarly for the second inequality. ∎

######  Proposition 4.35.

Under standard assumptions if $D_{k}$ are ${\varepsilon}$-appropriate sets for
$k\in K$, then $D=\bigcup_{k\in K}D_{k}$ is ${\varepsilon}$-appropriate.

###### Proof.

We have $\theta^{\mu}(D)=\theta^{\mu}(K)$ from Lemma 2.5. For the third
condition in Definition 4.16 we take $C_{U,i}^{D}=\sum_{k\in
K}C_{U,i}^{D_{k}}$. These are good approximations as
$\mu_{2}({\tilde{D}})=\sum_{k}\mu_{2}({\tilde{D}}_{k})$, and obviously satisfy
the proportion inequality. ∎

##  5 The $\Theta$ theorem

###  5.1 Preparations for divisibility

In this section we shall prove the main theorem concerning $\Theta$ functions.
Under standard assumptions, we shall consider $\mu$ to be a fixed proper
measure on ${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$.
By $\mu_{2}$ we shall denote the restriction of $\mu$ to
${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times\\{0\\}$. Note that as the
support of $\mu$ is a c-set with respect to the $n-2$ variables of
${\mathbb{R}}^{n-2}$, the support of $\mu_{2}$ is the projection of the
support of $\mu$. As we fix $\mu$, we shall omit the upper index when writing
the $\Theta$ function and write $\theta$ or $\theta_{k}$ instead of
$\theta^{\mu}$ or $\theta_{k}^{\mu}$.

The main theorem we want to prove is:

######  Theorem 5.1.

Under non-degenerate assumptions $\theta(A)\geq\theta(K)$ and
$\theta(K)\geq\theta({\bar{A}})$, whenever both sides of an inequality are
defined.

This looks like a quite simple theorem, and we suspect there is a simpler
proof than the one we present here. However, we were not able to find it (and
would be interested to learn if anyone does). Notice that if $\theta(A)$ is
undefined, then $\int_{A}\eta_{1}d\mu=0$, which implies
$\int_{A}\eta_{2}d\mu=0$ from property (T3). Thus
$\theta({\bar{A}})=\theta(K)$ and the Theorem is satisfied. Thus we assume
$\theta(A)$ is defined. Similarly we may assume $\theta({\bar{A}})$ is
defined. From Fact 2.2 it is enough to prove $\theta(A)\geq\theta(K)$ and the
second inequality will follow. Thus, we concentrate on the first inequality.
First, for technical reasons, we shall deal with the low-dimensional case:

######  Theorem 5.2.

Under standard assumptions with $n\leq 2$ (that is, $K\subset{\mathbb{R}}$ or
$K\subset{\mathbb{R}}^{2}$) we have $\theta(A)\geq\theta(K)$ and
$\theta(K)\geq\theta({\bar{A}})$, whenever both sides of an inequality are
defined.

###### Proof.

For $n=1$ the set $A$ is one-dimensional, and thus (being a c-set) is an
interval of the form $[0,a)$. We apply property (T4) to $K^{\prime}=K$, the
measure $\mu$ and the decomposition ${\mathbb{R}}\times\\{0\\}$ and get that
$\frac{\eta_{2}}{\eta_{1}}$ is a decreasing function. Thus from Lemma 2.3,
part 1a, $\theta(A)\geq\theta({\bar{A}})$ and the thesis follows from Fact
2.2.

For $n=2$ we shall approximate the set $A$ by a $l$-stair set. The $l$-stair
set is defined as follows:

######  Definition 5.3.

A $1$-stair set defined by $x_{1}=0$ and $a_{1}\geq 0$ (denoted
$A(x_{1};a_{1})$) is the empty set. A $l$-stair set defined by $0=x_{1}\leq
x_{2}\leq\ldots\leq x_{l}$ and $a_{1}\geq a_{2}\geq\ldots\geq a_{l}\geq 0$,
denoted $A(x_{1},x_{2},\ldots,x_{l};a_{1},a_{2},\ldots,a_{l})$ is defined by

$A(x_{1},x_{2},\ldots,x_{l};a_{1},a_{2},\ldots,a_{l})=\Big{(}A(x_{1},x_{2},\ldots,x_{l-1};a_{1},a_{2},\ldots,a_{l-1})\cap\\{x\leq
x_{l}\\}\Big{)}\cup A(0,a_{l}).$

That means that a $l$-stair set consists of $l$ steps, the $k$-th step goes
from $x_{k}$ to $x_{k+1}$ (the last one goes all the way to infinity) at
height $a_{k}$. A proper $l$-stair set is a $l$-stair set with $a_{l}=0$

Notice that $\theta(A)=\theta(K\cap A)$ as ${\rm supp}\mu\subset K$. Thus we
may assume $A\subset K$, and thus $A$ is bounded. Take
$A_{n}=\\{(x,y):([xn]/n,y)\in A\\}$, where $[xn]$ denotes the integer part of
$xn$. This is a proper stair set defined by $0,1/n,2/n,\ldots$ (a finite
sequence as $A$ is bounded) and the sequence $a_{k}=\sup\\{y:(k/n,y)\in A\\}$.
Notice also $A_{2^{n}}\supset A_{2^{n+1}}\supset A$ and
$\mu(A_{2^{n}}\setminus A){\rightarrow}0$. Thus
$\theta(A_{2^{n}}){\rightarrow}\theta(A)$, so it is enough to prove
$\theta(A_{2^{n}})\geq\theta(K)$ and go to the limit. Thus, instead of
considering all c-sets we may restrict ourselves to proper $l$-stair sets.

The proof for $A$ being a proper $l$-stair set will be an induction upon $l$.
For $l=1$ the set $A$ is empty and the thesis is obvious. For $l=2$ let
$I=[0,x_{2}]$. From Lemma 4.31 the function $\theta_{1}(y;I)$ is decreasing
where defined. Thus from Lemma 2.3 we have
$\theta(I\times[0,a_{1}])\geq\theta_{1}(a_{1};I)\geq\theta(I\times[a_{1},\infty))$.
Note that $A=I\times[0,a_{1}]$. Thus if $\theta(A)\geq\theta(K)$ or
$\theta(A)$ is undefined, the thesis is satisfied. Otherwise, as
$\theta(K)>\theta(I\times[0,a_{1}])$ we have
$\theta(K)>\theta(I\times[a_{1},\infty))$ if defined, and thus from Lemma 2.5
$\theta(K)>\theta(I\times{\mathbb{R}}_{y})$. Now apply property (T4) to
$K^{\prime}=K$, the decomposition ${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$ and
the measure $\mu$ to get that $\theta_{1}(x)$ is a decreasing function. Again
from Lemma 2.3 and Lemma 2.5 this implies
$\theta(I\times{\mathbb{R}}_{y})\geq\theta(K)$, a contradiction. Thus the
thesis is satisfied for $l=2$.

For larger $l$ let $I=[x_{l-1},x_{l}]$. Again from Lemma 4.31 the function
$\theta_{1}(y;I)$ is decreasing. Thus

$\theta(I\times[0,a_{l-1}])\geq\theta(I\times[a_{l-1},a_{l-2}])$ (5.1.1)

if both are defined. Note

$A(x_{1},x_{2},\ldots,x_{l};a_{1},a_{2},\ldots,a_{l})\setminus(I\times[0,a_{l-1}])=A(x_{1},x_{2},\ldots,x_{l-1};a_{1},a_{2},\ldots,a_{l-2},0)$

and

$A(x_{1},x_{2},\ldots,x_{l};a_{1},a_{2},\ldots,a_{l})\cup(I\times[a_{l-1},a_{l-2}])=A(x_{1},x_{2},\ldots,x_{l-2},x_{l};a_{1},a_{2},\ldots,a_{l-2},a_{l}).$

Suppose $\theta(A)<\theta(K)$. If $\theta(I\times[0,a_{l-1}])>\theta(A)$ or is
undefined, then from Lemma 2.5 we have
$\theta(A\setminus(I\times[0,a_{l-1}]))\leq\theta(A)$, but from the inductive
assumption for $l-1$ we have
$\theta(A\setminus(I\times[0,a_{l-1}]))\geq\theta(K)$, from which
$\theta(A)\geq\theta(K)$. If, on the other hand,
$\theta(I\times[0,a_{l-1}])\leq\theta(A)$, then from (5.1.1)
$\theta(I\times[a_{l-1},a_{l-2}])\leq\theta(A)$ or is undefined, thus from
Lemma 2.5 $\theta(A)\geq\theta(A\cup(I\times[a_{l-1},a_{l-2}]))$, and again
from the inductive assumption $\theta(A)\geq\theta(K)$.

Thus for any $l$ and for any $A$ being a proper $l$-stair set we have
$\theta(A)\geq\theta(K)$, which ends the proof. ∎

###### Proof of the Theorem 5.1.

The proof will proceed by induction upon $n$. For $n\leq 2$ we use Theorem
5.2.

For greater $n$ let
$K\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$.
Assume the thesis is true for all cases with $n^{\prime}<n$. If the theorem
holds for strict $\Theta$ functions, then for any non-degenerate $\theta$ we
take a sequence $\theta_{i}$ of strict $\Theta$ functions for
${\varepsilon}=1/i$. For any set $C$ for which $\theta(C)$ is defined, we have
$\theta_{i}(C){\rightarrow}\theta(C)$, so as we had
$\theta_{i}(A)\geq\theta_{i}(K_{i})\geq\theta_{i}({\bar{A}})$, we get the
thesis when $i$ tends to infinity. Thus it is enough to restrict ourselves to
strict assumptions. Note that under strict assumptions $\theta(U)$ is defined
for any set $U$ with $\mu(U)>0$ as ${\rm supp}\mu\subset{\rm supp}\eta_{1}$.
In particular, if $\mu_{2}({\tilde{U}})>0$, then
$\theta(U\times{\mathbb{R}}^{n-2})$ is well defined.

Also note that if $\theta(K_{+})=0$, then $\eta_{2}$ has to be zero
$\mu_{2}$-almost everywhere, which means $\theta(U)=0$ for any $U$ such that
it is defined, thus Theorem 5.1 holds. Thus we can assume $\theta(K_{+})>0$.

We want to prove that for any ${\varepsilon}>0$ the quadrant $K_{+}$ is an
${\varepsilon}$-appropriate set. We shall frequently require the following
property from various sets $D$:

$\theta(D)=\theta(K),$ (5.1.2)

or (for lower-dimensional sets)

$\theta_{k}(\mathbf{a};D)=\theta(K).$ (5.1.3)

We shall need to bound the diameter of the constructed sets from below. To
this end consider the following sets:
${\tilde{S}}^{0}=\\{(x,y):\theta_{n-2}(x,y)=\theta(K)\\}$,
${\tilde{S}}^{+}=\\{(x,y):\theta_{n-2}(x,y)\geq\theta(K)\\}$,
${\tilde{S}}^{-}=\\{(x,y):\theta_{n-2}(x,y)\leq\theta(K)\\}$ and
${\tilde{S}}={\rm cl}{\tilde{S}}^{+}\cap{\rm cl}{\tilde{S}}^{-}$. We take a
$\delta$-neighbourhood ${\tilde{S}}_{\delta}$ of ${\tilde{S}}$ with $\delta$
so small that

$\mu({\tilde{S}}_{\delta}\setminus{\tilde{S}})\leq{\varepsilon}\mu_{2}({\tilde{K}})(\lambda_{n-2}(K\cap\\{x=0,y=0\\})\sup_{K}\eta_{1})^{-1}/3.$

Note that as from property (T4) the function $\theta_{n-2}(x,y)$ is
coordinate-wise decreasing, the set ${\tilde{S}}\setminus{\tilde{S}}^{0}$ has
measure 0.

######  Remark 5.4.

Note that any ${\tilde{D}}\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$
having property (5.1.2) must, from Fact 2.5, have a non-empty intersection
both with ${\tilde{S}}^{+}$ and ${\tilde{S}}^{-}$ in some points where the
density of $\mu_{2}$ is positive. Thus any convex set ${\tilde{D}}$ with
property 5.1.2 will satisfy ${\tilde{D}}\cap{\rm
supp}\mu_{2}\cap{\tilde{S}}\neq\emptyset$, as the set ${\tilde{D}}\cap{\rm
supp}\mu_{2}$ is convex, and thus connected. Thus either ${\tilde{D}}\cap{\rm
supp}\mu_{2}$ is contained in ${\tilde{S}}_{\delta}$ or it has diameter at
least $\delta$.

The main part of the proof will be an transfinite inductive construction of
subsequent ${\varepsilon}$-appropriate strict lens sets by the following
Theorem:

######  Theorem 5.5.

Let $n>2$. Assume Theorem 5.1 holds under non-degenerate assumptions for any
$n^{\prime}<n$. Then under strict assumptions if $\theta(K_{+})>0$ for any
ordinal $\gamma$ there exists a division of the set $\tilde{K}_{+}$ into
$\gamma+2$ sets $U(\gamma,\beta)$ for $0\leq\beta\leq\gamma+1$ satisfying:

  * •

The set $U(\gamma,\gamma+1)$ is of $\mu_{2}$ measure at most
${\varepsilon}^{\prime}\mu_{2}({\tilde{K}})(\lambda_{n-2}(K\cap\\{x=0,y=0\\})\sup_{K}\eta_{1})^{-1}$.

  * •

The set $U(\gamma,\gamma)$ is either an appropriate set, a strict lens set
satisfying condition (5.1.2) or has $\mu_{2}$ measure 0.

  * •

All sets $U(\gamma,\beta)$ for $\beta<\gamma$ are either
${\varepsilon}^{\prime}$-appropriate sets, empty, or have non-zero $\mu_{2}$
measure and satisfy
$U(\gamma,\beta)\cap\tilde{K}_{+}\subset{\tilde{S}}_{\delta}$

  * •

If any $U(\gamma,\beta)$ is empty for $\beta<\gamma$, then $U(\gamma,\gamma)$
has measure 0.

If we prove this Theorem, we can apply it to prove Theorem 5.1. By the
inductive assumption we assume Theorem 5.1 holds for $n^{\prime}<n$. We take
$\gamma=\omega_{1}$ and ${\varepsilon}^{\prime}={\varepsilon}/3$. As the
measure of $\tilde{K}_{+}$ is finite, it cannot have $\omega_{1}$ disjoint
subsets of non-zero measure, thus some of $U(\omega_{1},\beta)$ for
$\beta<\omega_{1}$ are empty. Thus $U(\omega_{1},\omega_{1})$ has measure 0.

Let ${\tilde{T}}$ be the sum of those $U(\omega_{1},\beta)$ which are subsets
of ${\tilde{S}}_{\delta}$. For every point $(x,y)$ in
${\tilde{T}}\cap{\tilde{S}}^{0}$ we apply Theorem 5.1 to the restrictions of
$K,A,\theta$ and $\mu$ to $(x,y)\times{\mathbb{R}}^{n-2}$. The conditions are
satisfied — the restiction of $\theta$ is a derivative of $\theta$ and thus
non-degenerate, the restriction of $K$ is an generalized Orlicz ball due to
Lemma 4.19 and the restriction of $\mu$ is a proper measure due to 4.27, the
restriction of a c-set is obviously a c-set. Thus for all
$(x,y)\in{\tilde{S}}^{0}$ we have

$\theta_{n-2}(x,y;A)\geq\theta_{n-2}(x,y)\geq\theta_{n-2}(x,y;{\bar{A}}),$

and as $\theta_{n-2}(x,y)=\theta(K)$ from the definition of ${\tilde{S}}^{0}$,
from Lemma 2.5 we get

$\theta\big{(}(({\tilde{T}}\cap{\tilde{S}}^{0})\times{\mathbb{R}}^{n-2})\cap
A\big{)}\geq\theta(K)\geq\theta\big{(}(({\tilde{T}}\cap{\tilde{S}}^{0})\times{\mathbb{R}}^{n-2})\cap{\bar{A}}\big{)},$

and also
$\theta(({\tilde{T}}\cap{\tilde{S}}^{0})\times{\mathbb{R}}^{n-2})=\theta(K)$,
if only $\mu({\tilde{T}}\cap{\tilde{S}}^{0})>0$. Thus
${\tilde{T}}\cap{\tilde{S}}^{0}$ either has measure 0, or is an appropriate
set. Meanwhile ${\tilde{T}}\setminus{\tilde{S}}^{0}$ has measure at most
${\varepsilon}(\lambda_{n-2}(K\cap\\{x=0,y=0\\})\sup_{K}\eta_{1})^{-1}\mu_{2}({\tilde{K}})/3$
from the definition of ${\tilde{S}}_{\delta}$.

We therefore have a division of $\tilde{K}_{+}$ except a set of measure
$2{\varepsilon}(\lambda_{n-2}(K\cap\\{x=0,y=0\\})\sup_{K}\eta_{1})^{-1}\mu_{2}({\tilde{K}})/3$
into $({\varepsilon}/3)$-appropriate sets. The sum of all the
$({\varepsilon}/3)$-appropriate sets is by Remark 4.35 an
$({\varepsilon}/3)$-appropriate set. As the integral of $\eta_{i}$ over the
remaining set is at most $2{\varepsilon}\mu_{2}({\tilde{K}})/3$, the whole
$\tilde{K}_{+}$ is an ${\varepsilon}$-appropriate set with the same $C_{U,i}$.
As we can do this for any ${\varepsilon}>0$, by Lemma 4.34 $K$ is an
appropriate set, which is the thesis of Theorem 5.1 ∎

###  5.2 Almost horizontal divisions

We shall prove that if we can divide a lens set with a horizontal, or even
almost horizontal (under strict assumptions) line into two sets with equal
$\theta$, then the lens set is appropriate.

######  Lemma 5.6.

Assume Theorem 5.1 holds for $n^{\prime}<n$. Under strict assumptions if for a
given lens set $D\subset{\mathbb{R}}^{n}$ satisfying (5.1.2) there exists a
horizontal or vertical line $L$ in ${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$
dividing ${\tilde{D}}$ into two sets ${\tilde{D}}_{-}$ and ${\tilde{D}}_{+}$
of non-zero $\mu_{2}$-measure with
$\theta({\tilde{D}}_{-}\times{\mathbb{R}}^{n-2})=\theta({\tilde{D}}_{+}\times{\mathbb{R}}^{n-2})$,
then $D$ is an appropriate set.

###### Proof.

Suppose, without loss of generality, the line is horizontal given by
$y=a_{0}$. From Corollary 4.33 we know that for any $a$ we also have
$\theta_{n-1}(y=a;D)=\theta(D)$ if defined.

From Lemma 4.27 and property (T7) we know that the restriction of $\mu$ to
$\\{y=a\\}$ is a proper measure and the restriction of $\theta$ is a non-
degenerate $\Theta$ function. From the assumption we can apply Theorem 5.1,
thus

$\theta_{n-1}(y=a;D\cap
A)\geq\theta_{n-1}(y=a;D)\geq\theta_{n-1}(y=a;D\cap{\bar{A}}).$

As $\theta_{n-1}(y=a;D)=\theta(D)$, which does not depend on $a$, we can apply
Lemma 2.5 to get $\theta(D\cap A)\geq\theta(D)\geq\theta(D\cap{\bar{A}})$, and
as $\theta(D)=\theta(K)$ this means $D$ is appropriate. ∎

######  Lemma 5.7.

Assume Theorem 5.1 holds for $n^{\prime}<n$. Under strict assumptions if for a
given strict lens set $D$ satisfying (5.1.2) for every $\beta>0$ there exists
a line $L_{\beta}$ with inclination between $0$ and $\beta$ (i.e. almost
horizontal) or between $\frac{\pi}{2}-\beta$ and $\frac{\pi}{2}$ (i.e. almost
vertical) dividing ${\tilde{D}}$ into two sets ${\tilde{D}}_{-}$ and
${\tilde{D}}_{+}$ of non-zero $\mu_{2}$-measure with
$\theta(D_{-})=\theta(D_{+})=\theta(D)$, then $D$ is an appropriate set.

###### Proof.

Assume $\theta(D)>0$ (otherwise the thesis is trivial). We choose a sequence
of such lines $L_{i}$ with $\beta{\rightarrow}0$. We choose a subsequence such
that all lines are almost vertical or all are almost horizontal (we shall
assume without loss of generality that all are almost horizontal). From the
compactness of the set of lines intersecting the closure of
${\tilde{D}}\cap{\rm supp}(\mu_{2})$ we can choose a subsequence of lines
converging to some line $L$, which, of course, will be horizontal. If $L$ cuts
off a non-zero $\mu_{2}$ measure both above and below it, then both the sets
into which ${\tilde{D}}$ is divided have the same $\theta=\theta(D)$ from the
continuity of $\theta$ with respect to the set and the thesis follows from
Lemma 5.6. The case left to examine is when $L_{i}$ approaches the lowest or
highest point $p$ of ${\tilde{D}}\cap{\rm supp}(\mu_{2})$.

From the definition of a lens set we know that the only points of
${\tilde{D}}$ on which $\mu_{2}$ vanishes lie outside $\tilde{K}_{+}$. Thus
the lowest point of ${\tilde{D}}\cap{\rm supp}(\mu_{2})$ is the lower extremal
point of ${\tilde{D}}$. The highest point can be either the upper extremal
point of ${\tilde{D}}$ or can lie on the boundary of ${\rm supp}\mu_{2}$.

First consider the second, simpler case. As $D$ is a strict lens set and $K$
is proper, for any neighbourhood
${\tilde{U}}\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$ of the highest
point $p$ if we take a sufficiently horizontal line passing sufficiently close
to $p$, the set it will cut off from ${\tilde{D}}$ will be a subset of
${\tilde{U}}$ (${\tilde{D}}\cap{\rm supp}\mu_{2}$ has no horizontal edges). We
know that ${\rm supp}\eta_{2}\subset{\rm Int}\ {\rm supp}\mu$, so ${\rm cl}\
\widetilde{{\rm supp}\eta_{2}}\subset{\rm supp}\mu_{2}$. As $p$ lies on the
boundary of ${\rm supp}\mu_{2}$, it lies outside ${\rm Int}\ {\rm
supp}\mu_{2}$ and thus outside ${\rm cl}\ \widetilde{{\rm supp}\eta_{2}}$, so
we can choose an open neighbourhood ${\tilde{U}}$ of $p$ on which $\eta_{2}$
is 0. This neighbourhood has non-zero $\mu_{2}$ measure, and as
$\mu_{2}({\tilde{D}})>0$, $\mu_{2}({\tilde{D}}\cap{\tilde{U}})>0$. But
$\eta_{2}$ on the whole set ${\tilde{U}}$ is zero, thus any line cutting off
only a part of ${\tilde{U}}$ cannot satisfy $\theta(D_{+})=\theta(D)>0$.

In the first case $(L_{i})$ approaches one of the extremal points of
${\tilde{D}}$. Assume it is the lower point. For any line $L_{i}$ the set
${\tilde{D}}^{L_{i}}_{-}$ is a lens set. From Lemma 2.5 there has to be a
point $p_{i}\in D^{L_{i}}_{-}$ with
$\theta(\\{p_{i}\\}\times{\mathbb{R}}^{n-2})\leq\theta(D^{L_{i}}_{-})=\theta(D)$.
The lines $L_{i}$ tend to the horizontal line through $(x_{1},y_{1})$, the
lower extremal point of $D$. Thus, the vertical coordinate of $p_{i}$ tends to
$y_{1}$, and as ${\tilde{D}}$ has no horizontal edges, the horizontal
coordinate of $p_{i}$ tends to $x_{1}$, meaning
$p_{i}{\rightarrow}(x_{1},y_{1})$.

From property (T4), as
$\theta_{n-2}(\\{p_{i}\\}\times{\mathbb{R}}^{n-2})\leq\theta(D)$, for all
points $p\in D$ except for $(x_{1},y_{1})$ we have
$\theta_{n-2}(\\{p\\}\times{\mathbb{R}}^{n-2})\leq\theta(D)$. This, however,
from Lemma 2.5 implies in particular, that for any horizontal line $M$
dividing ${\tilde{D}}$ into two sets of non-zero $\mu$-measure we have
$\theta(D_{+})\leq\theta(D)$, which from Lemma 4.33 implies
$\theta(D_{+})=\theta(D)$, which from Lemma 5.6 implies that $D$ is
appropriate. ∎

###  5.3 ${\varepsilon}$-appropriateness of lens sets

This subsection puts down precisely what we meant by “long and narrow” in the
idea of the proof, and show how to go from the “longness and narrowness” to
${\varepsilon}$-appropriateness.

######  Lemma 5.8.

Let $C\subset{\mathbb{R}}^{n}$ be a convex set with $\lambda(C)>0$, let
$I\subset C$ be an interval of length $a$, and let $L$ be the line containing
$I$. Let $f:C{\rightarrow}(0,\infty)$ be a $1/m$-concave function. Let
$P:{\mathbb{R}}^{m}{\rightarrow}L$ be the orthogonal projection onto $L$. Let
$J\subset I$ be an subinterval of length $b$. Let $C^{\prime}\subset C$ be
such a set that $P(C^{\prime})\subset J$. Then

$\int_{C^{\prime}}f(x)dx\leq\Big{(}\Big{(}\frac{a+b}{a-b}\Big{)}^{n+m}-1\Big{)}\int_{C}f(x)dx$

and also

$\int_{C^{\prime}}f(x)dx\leq\frac{2^{n+m+2}b}{a}\int_{C}f(x)dx.$

###### Proof.

Let $p(y)=\int_{x:P(x)=y}f(x)dx$ and let $I^{\prime}=\\{y\in L:p(y)>0\\}$.
From Fact 4.24 the function $p(y)=\int_{x:P(x)=y}f(x)dx$ is a
$(1/m+n-1)$-concave function on $L$, thus $I^{\prime}$ is an interval. As $f$
is positive and $C$ is convex and has positive measure, $p$ is positive on
${\rm Int}I$, thus the length of $I^{\prime}$ is at least $a$. If $J\cap
I^{\prime}=\emptyset$, then $\int_{C^{\prime}}f(x)dx=0$ and the thesis is
satisfied, so assume $J\cap I^{\prime}\neq\emptyset$. Then
$I^{\prime}\setminus J$ is a sum of two intervals (one may be empty) of total
length at least $a-b$. Thus it contains an interval $I^{\prime\prime}$ of
length at least $\frac{a-b}{2}$, let $\\{y_{1}\\}={\rm
cl}I^{\prime\prime}\cap{\rm cl}J$ and $y_{2}$ be the other end of
$I^{\prime\prime}$. Let $y_{2}$ and $y_{3}$ be the ends of $I^{\prime}$ and
let $T$ be such that $y_{3}=Ty_{1}+(1-T)y_{2}$ (as $y_{1}$ lies between
$y_{2}$ and $y_{3}$ we know $T\geq 1$).

As $p$ is $1/(n+m-1)$-concave,

$p^{1/(n+m-1)}(ty_{1}+(1-t)y_{2})\geq
tp^{1/(n+m-1)}(y_{1})+(1-t)p^{1/(n+m-1)}(y_{2})\geq tp^{1/(n+m-1)}(y_{1})$

for $t\in[0,1]$, which means

$\int_{I^{\prime\prime}}p(y)dy\geq|I^{\prime\prime}|\int_{[0,1]}t^{n+m-1}p(y_{1})dt=|I^{\prime\prime}|\frac{1}{n+m}p(y_{1}).$

Similarly for $T\geq t\geq 1$ we have

$p^{1/(n+m-1)}(ty_{1}+(1-t)y_{2})\leq
tp^{1/(n+m-1)}(y_{1})+(1-t)p^{1/(n+m-1)}(y_{2})\leq tp^{1/(n+m-1)}(y_{1})$

for $t\in[1,T]$, which gives

$\int_{J}p(y)\leq|I^{\prime\prime}|\int_{1}^{(|J|+|I^{\prime\prime}|)/|I^{\prime\prime}|}t^{n+m-1}p(y_{1})=|I^{\prime\prime}|\frac{1}{n+m}\Big{(}\Big{(}\frac{a+b}{a-b}\Big{)}^{n+m}-1\Big{)}p(y_{1}).$

Thus

$\frac{\int_{C^{\prime}}f(x)dx}{\int_{C}f(x)dx}\leq\Big{(}\frac{a+b}{a-b}\Big{)}^{n+m}-1,$

which proves the first part of the Lemma.

For the second part note that if $a/b\leq 2^{m+n+2}$, then the thesis is true,
as $\int_{C^{\prime}}f(x)dx\leq\int_{C}f(x)dx$ because $C^{\prime}\subset C$.
For $b/a\leq 2^{-(n+m+1)}$ we have

$\Big{(}\frac{a+b}{a-b}\Big{)}^{n+m}-1=\Big{(}\frac{1+b/a}{1-b/a}\Big{)}^{n+m}-1\leq\frac{1+2^{n+m}b/a}{1-2^{n+m}b/a}-1\leq\frac{2^{n+m+1}b/a}{1/2}=2^{n+m+2}b/a.$

∎

######  Corollary 5.9.

Let ${\varepsilon}>0$. Let $\mu$ be a measure on ${\mathbb{R}}^{2}$ with a
$1/m$ concave density. Let ${\tilde{D}}\subset{\mathbb{R}}^{2}$ be a lens set.
Let $L$ be the extremal line of ${\tilde{D}}$ and
$p:{\mathbb{R}}^{2}{\rightarrow}L$ the orthogonal projection onto $L$. Let $A$
be a c-set in ${\mathbb{R}}^{2}$. Let $A^{\prime}=p^{-1}(A\cap L)$. Assume the
relevant length of ${\tilde{D}}$ (that is, the length of
$L\cap{\tilde{D}}\cap{\rm supp}\mu$) is at least $d>0$, the inclination of
${\tilde{D}}$ between $\beta$ and $\pi/2-\beta$ with $\beta>0$ and width at
most $w=\frac{1}{2\max(\cot\beta,\tan\beta)}2^{-m-3}{\varepsilon}d$. Then
$\mu((A\bigtriangleup
A^{\prime})\cap{\tilde{D}})\leq{\varepsilon}\mu({\tilde{D}})$.

###### Proof.

Let $p=(x_{p},y_{p})$ be the rightmost point on $L\cap A$ (and at the same
time on $L\cap A^{\prime}$, from the definition of $A^{\prime}$). As both $A$
and $A^{\prime}$ are c-sets, we have $A\bigtriangleup
A^{\prime}\subset\\{(x,y):x>x_{p},y<y_{p}\\}\cup\\{(x,y):x<x_{p},y>y_{p}\\}$.
As $D$ has width at most $w$, the projection of $(A\bigtriangleup
A^{\prime})\cap D$ onto $L$ has length at most $2w\max(\tan\beta,\cot\beta)$.
From Lemma 5.8 we know that as
$2w\max(\tan\beta,\cot\beta)<2^{-m-3}{\varepsilon}d$, we have
$\mu((A\bigtriangleup A^{\prime})\cap D)\leq{\varepsilon}\mu(D)$. ∎

######  Corollary 5.10.

Let $K\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$
be a generalized Orlicz ball with a proper measure $\mu$ and $D$ be a lens set
of relevant length at least $d$, inclination between $\beta$ and $\pi/2-\beta$
and width at most
$w=\frac{1}{2\max(\cot\beta,\tan\beta)}2^{-m-3}{\varepsilon}d(\lambda_{n-2}(K\cap\\{x=0,y=0\\}))^{-1}$,
where $m$ is such that the density of $\mu$ is $1/m$ concave. Let $A$ be a
c-set in ${\mathbb{R}}^{n}$ and let $A^{\prime}$ be defined as before. Then

$\mu((A\bigtriangleup A^{\prime})\cap
D)\leq{\varepsilon}\mu_{2}({\tilde{D}}).$

###### Proof.

For each $t\in{\mathbb{R}}^{n-2}$ we may apply Corollary 5.9, and integrate
over $K\cap\\{x=0,y=0\\}$ to get a bound for the Lebesgue measure. ∎

Note the same argument works if $A$ is the complement of a c-set.

######  Corollary 5.11.

Let $K\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$
be a generalized Orlicz ball with a proper measure $\mu$. Let $D$ be a lens
set of relevant length at least $d$, inclination between $\beta$ and
$\pi/2-\beta$ and width at most
$w=\frac{1}{2\max(\cot\beta,\tan\beta)}2^{-m-3}dM^{-1}{\varepsilon}(\lambda_{n-2}(K\cap\\{x=0,y=0\\}))^{-1}$
and $\phi:{\mathbb{R}}^{n}{\rightarrow}[0,M]$ be a coordinate-wise decreasing
function with $\bar{\phi}(t):=\phi(p(t))$, where $p$ is the orthogonal
projection onto $L\times{\mathbb{R}}^{n-2}$, $L$ being the extremal line of
${\tilde{D}}$. Then for any $U\subset D$ we have

$\big{|}\int_{U}\phi(t)d\mu(t)-\int_{U}\bar{\phi}(t)d\mu(t)\big{|}<{\varepsilon}\mu_{2}({\tilde{D}}).$

###### Proof.

As $\phi$ is coordinate-wise decreasing, the sets $\phi^{-1}([s,\infty))$ are
c-sets. By the integration by parts,

$\int_{U}\phi(t)d\mu(t)=\int_{0}^{M}\mu(\phi^{-1}([s,\infty)\cap U)ds.$

The sets $(\bar{\phi})^{-1}([s,\infty))$ are formed from the sets
$\phi^{-1}([s,\infty))$ as in Corollary 5.10. Thus for each $s$ we have

$\displaystyle\Bigg{|}\mu\Big{(}\phi^{-1}([s,\infty)\cap
U\Big{)}-\mu\Big{(}(\bar{\phi})^{-1}([s,\infty)\cap U\Big{)}\Bigg{|}$
$\displaystyle\leq\mu\Big{(}\big{(}\phi^{-1}([s,\infty)\cap
U\big{)}\bigtriangleup\big{(}(\bar{\phi})^{-1}([s,\infty)\cap U\big{)}\Big{)}$
$\displaystyle\leq\mu\Big{(}\big{(}\phi^{-1}([s,\infty)\cap
D\big{)}\bigtriangleup\big{(}(\bar{\phi})^{-1}([s,\infty)\cap
D\big{)}\Big{)}\leq M^{-1}{\varepsilon}\mu_{2}({\tilde{D}}),$

which integrated over $[0,M]$ gives the thesis. ∎

######  Lemma 5.12.

Consider a generalized Orlicz ball
$K\subset{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}\times{\mathbb{R}}^{n-2}$ with
a proper measure $\mu$ with both its defining functions $1/m$ concave, a
strict non-degenerate $\Theta$ function, any ${\varepsilon}>0$ and any c-set
$A$. Assume Theorem 5.1 holds for $n^{\prime}<n$. Let $D$ be a lens set
satisfying $\theta(D)=\theta(K)$ of relevant length at least $\delta$,
inclination between $\beta$ and $\frac{\pi}{2}-\beta$ and width at most

$w=\frac{1}{2\max(\cot\beta,\tan\beta)}2^{-3m-4}d\min\\{1,(\sup_{K}\eta_{1})^{-1}\\}{\varepsilon}(\lambda_{n-2}(K\cap\\{x=0,y=0\\}))^{-1}.$

Then $D$ is an $8{\varepsilon}$-appropriate set.

###### Proof.

Let $L$ be the extremal line of $D$. We switch coordinates in the plane
${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$ to orthogonal coordinates $(u,v)$
such that $L=\\{v=0\\}$ and $u>0$ on the positive quadrant of
${\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$. Define for any set
$U\in\\{A,{\bar{A}}\\}$ the set $U^{\prime}$ by $U\cap\\{v=0\\}$ and
$U^{\prime\prime}$ by $U\times{\mathbb{R}}_{v}$. Let $K^{\prime}$ be a
generalized Orlicz ball in ${\mathbb{R}}_{u}\times{\mathbb{R}}^{n-2}$ such
that $K_{+}^{\prime}=K_{+}\cap\\{v=0\\}$ given by Lemma 4.19 and
$K^{\prime\prime}=K^{\prime}\times{\mathbb{R}}_{v}$. Let $\eta_{i}^{\prime}$
be the restriction of $\eta_{i}$ to $\\{v=0\\}$ and
$\eta_{i}^{\prime\prime}(u,v,t)=\eta_{i}(u,0,t)$. Let $\mu^{\prime}$ be the
measure on $K^{\prime}$ with density
$h(u)=\int_{{\mathbb{R}}_{v}}{\mathbf{1}}_{(u,v)\in{\tilde{D}}}f(u,v)g(u,v)$,
where $f$ and $g$ are the density functions defining $\mu$, and
$\mu^{\prime\prime}$ be the measure on ${\mathbb{R}}^{n}$ with density
$f(x)g(y)$ (without restricting to $K$).

We want to prove that $\int_{U^{\prime}\cap
K^{\prime}}\eta_{i}^{\prime}d\mu^{\prime}$ is a good approximation of
$\int_{U\cap D}\eta_{i}d\mu$, then check the assumptions for Theorem 5.1 on
$K^{\prime}$ and apply it for $A^{\prime}$. First note that

$\int_{K^{\prime}}\phi(u,t)d\mu^{\prime}(u,t)=\int_{K^{\prime\prime}\cap
D}\phi(u,0,t)d\mu^{\prime\prime}(u,v,t)$

for any function $\phi$ defined on $K^{\prime}$. This follows directly from
the definitions of $K^{\prime\prime}$, $\mu^{\prime}$ and
$\mu^{\prime\prime}$.

Let $M=\min\\{1,(\sup_{K}\eta_{1})^{-1}\\}$. We know $K_{+}$ is a c-set, thus
$\mu((K\bigtriangleup K^{\prime\prime})\cap
D)\leq{\varepsilon}\mu_{2}({\tilde{D}})$ by Corollary 5.10. Thus for any
$\phi$ we have

$\displaystyle\Bigg{|}\int_{K^{\prime}}\phi(u,t)d\mu^{\prime}(u,t)$
$\displaystyle-\int_{K\cap D}\phi(u,0,t)d\mu(u,v,t)\Bigg{|}=$
$\displaystyle\Bigg{|}\int_{K^{\prime\prime}\cap
D}\phi(u,t)d\mu^{\prime\prime}(u,v,t)-\int_{K\cap
D}\phi(u,0,t)d\mu^{\prime\prime}(u,v,t)\Bigg{|}<M{\varepsilon}\mu_{2}({\tilde{D}})\sup|\phi|.$

We repeat the same trick for
$U\in\\{A^{\prime\prime},{\bar{A}}^{\prime\prime}\\}$, putting
$\phi^{\prime}=\phi\cdot{\mathbf{1}}_{U}$ in the above inequality and applying
Corollary 5.10 again to get

$\displaystyle\Bigg{|}\int_{K^{\prime}\cap
A^{\prime}}\phi(u,t)d\mu^{\prime}(u,t)-\int_{K\cap D\cap
A}\phi(u,0,t)d\mu(u,v,t)\Bigg{|}<M{\varepsilon}\mu_{2}({\tilde{D}})\sup|\phi|.$

Finally, we insert $\eta_{i}$ for $\phi$ and apply Corollary 5.11 to get

$\Bigg{|}\int_{K^{\prime}\cap
A^{\prime}}\eta_{i}^{\prime}(u,t)d\mu^{\prime}(u,t)-\int_{K\cap D\cap
A}\eta_{i}(u,v,t)d\mu(u,v,t)\Bigg{|}\leq 3{\varepsilon}\mu_{2}({\tilde{D}}),$

and the same for integration over $K\cap D\cap{\bar{A}}$ and $K\cap D$.

Now we want to check assumptions for Theorem 5.1. $K^{\prime}$ is a
generalized Orlicz ball due to Lemma 4.19. $A^{\prime}$ is a c-set in
${\mathbb{R}}_{u}\times{\mathbb{R}}^{n-2}$ because $L$ is positively inclined,
thus an increase in $u$ translates to an increase in both $x$ and $y$.
$\mu^{\prime}$ is a projection of the measure with the density
$f(x)g(y){\mathbf{1}}_{D}$. The first two functions are $1/m$ concave, the
third is $1/1$ concave as $D$ is convex. Thus from Facts 4.23 and 4.24 the
density $h(u)$ of $\mu^{\prime}$ is a $1/(3m+1)$ concave function. Thus
$\mu^{\prime}$ is a proper measure on $K^{\prime}$ (recall $\mu^{\prime}$ is
restricted to $K^{\prime}$, thus the third point of the Definition 4.2 is
satisfied). $\eta_{1}^{\prime}$ and $\eta_{2}^{\prime}$ are restrictions of
$\eta_{1}$ and $\eta_{2}$ to $K^{\prime}$, which is a derivative of $K$, thus
they define a non-degenerate $\Theta$-function on $K^{\prime}$.

Let us apply Theorem 5.1. We get

$\frac{\int_{K^{\prime}\cap
A^{\prime}}\eta_{2}^{\prime}(u,t)d\mu^{\prime}(u,t)}{\int_{K^{\prime}\cap
A^{\prime}}\eta_{1}^{\prime}(u,t)d\mu^{\prime}(u,t)}\geq\frac{\int_{K^{\prime}}\eta_{2}^{\prime}(u,t)d\mu^{\prime}(u,t)}{\int_{K^{\prime}}\eta_{1}^{\prime}(u,t)d\mu^{\prime}(u,t)}\geq\frac{\int_{K^{\prime}\cap{\bar{A}}^{\prime}}\eta_{2}^{\prime}(u,t)d\mu^{\prime}(u,t)}{\int_{K^{\prime}\cap{\bar{A}}^{\prime}}\eta_{1}^{\prime}(u,t)d\mu^{\prime}(u,t)}.$
(5.3.1)

We need to make the middle expression equal to $\theta(D)$, so for any
$u_{0},t_{0}$ we define

$\bar{\eta}_{i}^{\prime}(u_{0},t_{0})=\frac{\int_{K\cap
D}\eta_{i}(u,v,t)d\mu(u,v,t)}{\int_{K^{\prime}}\eta_{i}^{\prime}(u,t)d\mu^{\prime}(u,t)}\eta_{i}^{\prime}(u_{0},t_{0}).$

As $\bar{\eta}_{i}^{\prime}=C_{i}\eta_{i}^{\prime}$, we have inequalities
(5.3.1) for functions $\bar{\eta}_{i}^{\prime}$ (although they do not
necessarily define a $\Theta$ function on $K^{\prime}$). To bound the error we
have

$\displaystyle\int_{K^{\prime}}\big{|}\bar{\eta}_{i}^{\prime}(u,t)-\eta_{i}^{\prime}(u,t)\big{|}d\mu^{\prime}(u,t)$
$\displaystyle=\int_{K^{\prime}}\eta_{i}^{\prime}(u,t)\Big{|}\frac{\int_{K\cap
D}\eta_{i}(u,v,t)d\mu(u,v,t)}{\int_{K^{\prime}}\eta_{i}^{\prime}(u,t)d\mu^{\prime}(u,t)}-1\Big{|}d\mu^{\prime}(u,t)$
$\displaystyle=\Bigg{|}\int_{K\cap
D}\eta_{i}(u,v,t)d\mu(u,v,t)-\int_{K^{\prime}}\eta_{i}^{\prime}(u,t)d\mu^{\prime}(u,t)\Bigg{|}\leq
3{\varepsilon}\mu_{2}({\tilde{D}}).$

As we bounded the integral of errors, the error on $K^{\prime}\cap A^{\prime}$
and $K^{\prime}\cap{\bar{A}}^{\prime}$ is no larger than
$3{\varepsilon}\mu_{2}({\tilde{D}})$.

We can now for $U\in\\{A^{\prime},{\bar{A}}^{\prime}\\}$ and $i\in\\{1,2\\}$
put $C_{U,i}=\int_{K^{\prime}\cap U}\bar{\eta}_{i}^{\prime}d\mu^{\prime}$.
Applying inequalities (5.3.1) to $\bar{\eta}_{i}^{\prime}$ we get

$\frac{C_{A,2}}{C_{A,1}}\geq\frac{\int_{K^{\prime}}{\bar{\eta}}_{2}^{\prime}(u,t)d\mu^{\prime}(u,t)}{\int_{K^{\prime}}\bar{\eta}_{1}^{\prime}(u,t)d\mu^{\prime}(u,t)}=\frac{\int_{K\cap
D}\eta_{2}(u,v,t)d\mu(u,v,t)}{\int_{K\cap
D}\eta_{1}(u,v,t)d\mu(u,v,t)}=\theta(D)=\theta(K)\geq\frac{C_{{\bar{A}},2}}{C_{{\bar{A}},1}},$

and putting together all the estimates we made we get $|C_{U,i}-\int_{K\cap
D\cap U}\eta_{i}d\mu|\leq 6{\varepsilon}\mu_{2}({\tilde{D}})$. ∎

##  6 The transfinite induction

What is left to prove is Theorem 5.5. We will prove by transfinite induction
an extended version of Theorem 5.5, which will allow us to carry the
information we need through the induction steps. The sets $U(\gamma,\beta)$
will have to satisfy the conditions of Theorem 5.5, and furthermore the
following conditions:

  * •

For any $\gamma>\beta$ we have $U(\gamma,\beta)=U(\beta+1,\beta)$.

  * •

For any $\gamma$ we have $U(\gamma,\gamma+1)=U(0,1)$.

  * •

If $\gamma$ is a successor ordinal and $U(\gamma,\gamma)$ has positive
$\mu_{2}$ measure, the sets $U(\gamma,\gamma-1)$ and $U(\gamma,\gamma)$ are
formed by dividing $U(\gamma-1,\gamma-1)$ with a straight line of positive
inclination.

  * •

If $\gamma$ is a limit ordinal,
$U(\gamma,\gamma)=\bigcap_{\beta<\gamma}U(\beta,\beta)$.

  * •

For any $\gamma$ if $U(\gamma,\gamma)$ has positive $\mu_{2}$ measure, then
for all $\beta<\gamma$ the sets $U(\beta,\beta)$ are strict lens sets.

Remark that this in fact means we carry out a transfinite inductive
construction. The sets $U(\gamma,\beta)$ for $\beta<\gamma$ depend only on the
second argument, once constructed. The set $U(\gamma,\gamma+1)$ is equal to
$U(0,1)$. The set $U(\gamma,\gamma)$ in each step has a part cut off to make a
new set $U(\gamma+1,\gamma+1)$.

Note that if $\theta(K)=0$, then $K$ is appropriate (as any $U\subset K$ with
$\mu_{2}(U)>0$ has $\theta(U)=0$). Thus by putting $U(\gamma,0)=\tilde{K}_{+}$
for any $\gamma$ and $U(\gamma,\beta)=\emptyset$ for $\gamma+1\geq\beta>0$ we
satisfy the conditions of Theorem 5.5. Thus, further on, we assume
$\theta(K)>0$.

###  6.1 Starting the transfinite induction

First we need to define the sets $U(0,0)$ and $U(0,1)$ to start the induction.
If we take $D=[x_{-},x_{+}]\times[y_{-},y_{+}]\times{\mathbb{R}}^{n-2}$, then
$D$ is a lens set and satisfies condition (5.1.2). It is not, however, a
strict lens set.

The idea is to take two almost vertical lines — one close to the left edge of
${\tilde{D}}$ and the other close to the right edge, then look at the $\theta$
of the set they cut off. If $\theta$ is too large, we move the left line
closer to the edge, if too small, we move the right line closer to the edge.
When we have balanced $\theta$, we repeat the same for horizontal lines. By
cutting off a bit from each edge we shall also ensure
$[x_{1},x_{2}]\subset(x_{-},x_{+})$ and similarly for $y$. Below is a
formalization of the argument.

If $\tilde{K}_{+}$ is appropriate to begin with, we take $U(0,1)=\emptyset$
and $U(0,0)=\tilde{K}_{+}$. Thus we assume $\tilde{K}_{+}$ is not appropriate.

Denote by $L^{-}(x,\beta)$ the line through $(x,y_{-})$ with inclination
$\pi/2-\beta$ and by $L^{+}(x,\beta)$ the line through $(x,y_{+})$ with
inclination $\pi/2-\beta$. Denote by ${\tilde{D}}^{-}(x,\beta)$ the subset of
$[x_{-},x_{+}]\times[y_{-}\times y_{+}]$ to the left of $L^{-}(x,\beta)$ and
by ${\tilde{D}}^{+}(x,\beta)$ the subset to the right of $L^{+}(x,\beta)$.
Note that for $\beta\in(0,\pi/2)$ those sets have positive $\mu_{2}$ measure
by the definition of a proper measure. Let
$\phi^{-}(x,\beta)=\theta({\tilde{D}}^{-}(x,\beta)\times{\mathbb{R}}^{n-2})-\theta(K)$
and
$\phi^{+}(x,\beta)=\theta({\tilde{D}}^{+}(x,\beta)\times{\mathbb{R}}^{n-2})-\theta(K)$.
From property T5 these functions are continuous in both arguments. From Lemma
4.33 and Lemma 5.7 there is a $\beta_{0}>0$ such that for $\beta<\beta_{0}$ we
have $\phi^{-}(x,\beta)>0$ and $\phi^{+}(x,\beta)<0$ for $x\in(x_{-},x_{+})$.

Now start with any $x_{l}$, $x_{u}$ and $0<\beta_{l},\beta_{u}<\beta_{0}$ such
that the sets ${\tilde{D}}^{-}(x_{l},\beta_{l})$ and
${\tilde{D}}^{+}(x_{u},\beta_{l})$ have measure no larger than
${\varepsilon}^{\prime}(\lambda_{n-2}(K\cap\\{x=0,y=0\\})\sup_{K}\eta_{1})^{-1}\mu(K)/4$
and do not intersect. Now if we fix $x_{u}$ and $\beta_{u}$ while letting
$x_{l}$ tend to $x_{-}$ and $\beta_{l}$ to 0, then $\theta$ of the sum of the
two sets will tend to
$\theta({\tilde{D}}^{+}(x_{u},\beta_{u})\times{\mathbb{R}}^{n-2})$, which is
strictly smaller than $\theta(K)$. If, on the other hand, we fix $x_{l}$ and
$\beta_{l}$ and let $x_{u}$ tend to $x_{+}$ and $\beta_{u}$ to 0, the $\theta$
of the two sets will approach
$\theta({\tilde{D}}^{-}(x_{l},\beta_{l})\times{\mathbb{R}}^{n-2})$, which is
strictly greater than $\theta(K)$. Thus, from the Darboux property, for some
values $x_{-}<x_{l}<x_{u}<x_{+}$ and $\beta_{l}$ and $\beta_{u}$ we have the
function

$\theta\Big{(}{\tilde{D}}^{+}(x_{u},\beta_{u})\times{\mathbb{R}}^{n-2}\cup{\tilde{D}}^{-}(x_{l},\beta_{l})\times{\mathbb{R}}^{n-2}\Big{)}=\theta(K).$

The set that remains is a lens set with no vertical boundaries and satisfies
property (5.1.2). If it is appropriate, we have found our $U(0,0)$ and define
$U(0,1)={\tilde{D}}^{-}(x_{l},\beta_{l})\cup{\tilde{D}}^{+}(x_{u},\beta_{u})$.
If not, then we can repeat the same trick for $y$ (we needed the non-
appropriateness to use Lemma 5.6), and achieve a lens set with no horizontal
and no vertical boundaries and separated from $x_{-}$ and $x_{+}$, i.e. a
strict lens set.

Thus we define
$U(0,1)={\tilde{D}}^{-}(x_{l},\beta_{l})\cup{\tilde{D}}^{+}(x_{u},\beta_{u})\cup{\tilde{D}}^{-}(y_{l},\alpha_{l})\cup{\tilde{D}}^{+}(y_{u},\alpha_{u})$
and and $U(0,0)=([x_{-},x_{+}]\times[y_{-},y_{+}])\setminus U(0,1)$.

######  Remark 6.1.

Assume $U(0,0)$ is a strict lens set (otherwise the induction will be
trivial). Recall $f$ and $g$ are $1/m$-concave functions defining the proper
measure $\mu$. As $U(0,0)$ is a strict lens set, it is separated from the
boundary of the support of $f\cdot g$. Thus (as $f$ and $g$ are continuous on
the interior of their support), they both attain positive minimal values
$f_{L}$ and $g_{L}$. Also, as they are continuous on their support and $1/m$
concave, they are bounded from above by some $f_{U}$ and $g_{U}$. Thus for any
set $T\subset U(0,0)$ we have

$f_{U}g_{U}\lambda_{2}(T)\geq\mu_{2}(T)\geq f_{L}g_{L}\lambda_{2}(T),$

and for any function $t$ on $T$ we have

$f_{U}g_{U}\int_{T}t(p)d\lambda_{2}(p)\geq\int_{T}t(p)d\mu_{2}(p)\geq
f_{L}g_{L}\int_{T}t(p)d\mu_{2}(p).$

###  6.2 The induction step for successor ordinals

For a successor ordinal $\gamma+1$ we have a division of $\tilde{K}_{+}$ for
$\gamma$. We put $U(\gamma+1,\gamma+2)=U(\gamma,\gamma+1)$. If
$U(\gamma,\gamma)$ is appropriate of positive measure, we put
$U(\gamma+1,\gamma)=U(\gamma,\gamma)$ (as an appropriate set is an
${\varepsilon}$-appropriate set) and $U(\gamma+1,\gamma+1)=\emptyset$. If
$U(\gamma,\gamma)$ has measure 0, we put $U(\gamma+1,\gamma)=\emptyset$ and
$U(\gamma+1,\gamma+1)=U(\gamma,\gamma)$. The difficult case to deal with will
be when $U(\gamma,\gamma)$ is a non-appropriate strict lens set. For brevity
denote $U(\gamma,\gamma)$ by ${\tilde{D}}$.

In this case from Lemma 5.7 there exists an angle $\alpha^{\prime}>0$ such
that any positively inclinated line dividing ${\tilde{D}}$ into two sets of
non-zero $\mu$-measure with equal $\theta$ has inclination greater than
$\alpha^{\prime}$ and smaller than $\frac{\pi}{2}-\alpha^{\prime}$. If the
inclination of ${\tilde{D}}$ is $\alpha^{\prime\prime}$, let
$\alpha=\min\\{\alpha^{\prime},\alpha^{\prime\prime},\frac{\pi}{2}-\alpha^{\prime\prime}\\}$.

We shall attempt to cut off a “long and narrow” lens set $U(\gamma+1,\gamma)$
from $U(\gamma,\gamma)$. We shall cut off a narrow set satisfying (5.1.2).
From Remark 5.4 it will either be long, or be a subset of
${\tilde{S}}_{\delta}$, both of which satisfy us.

Take a sufficiently small $w$
($w<\frac{1}{2\max(\cot\alpha,\tan\alpha)}2^{-3m-4}\delta\min\\{1,(\sup_{K}\eta_{1})^{-1}\\}\frac{{\varepsilon}}{8}(\lambda_{n-2}(K\cap\\{x=0,y=0\\})^{-1}$,
where $m$ is such that the density functions of $\mu$ are $1/m$-concave, will
suffice). For any angle $\xi\in[0,\frac{\pi}{2}]$ we can find continuously a
line $L_{\xi}$ of inclination $\xi$ such that the part ${\tilde{D}}_{+}(\xi)$
of ${\tilde{D}}\cap{\rm supp}\mu$ lying above and to the left of $L_{\xi}$ has
width no larger than $w$. From Lemma 4.33 we have
$\theta(D_{+}(0))\geq\theta(D)$ and $\theta(D_{+}(\pi/2))\leq\theta(D)$. From
the Darboux property for some $\xi$ we have $\theta(D_{+}(\xi))=\theta(D)$. We
take $U(\gamma+1,\gamma)={\tilde{D}}_{+}(\xi)$. Let $I_{\xi}$ denote the
segment of $L_{\xi}$ intersecting ${\tilde{D}}$.

The set $U(\gamma+1,\gamma+1)=U(\gamma,\gamma)\setminus U(\gamma+1,\gamma)$
is, of course, a strict lens set, satisfying condition (5.1.2), because the
new edge has inclination between $\alpha$ and $\frac{\pi}{2}-\alpha$, and all
the other edges come from the old set ${\tilde{D}}$. It remains to check that
$U(\gamma+1,\gamma)$ satisfies the conditions of the transfinite induction.
First let us check what is the inclination of $U(\gamma+1,\gamma)$. If both
the ends $I_{\xi}$ fall upon the upper-left border of $U(\gamma,\gamma)$, then
they are the extremal points of $U(\gamma+1,\gamma)$, and thus the inclination
of $U(\gamma+1,\gamma)$ is the inclination of the segment, which is between
$\alpha^{\prime}$ and $\frac{\pi}{2}-\alpha^{\prime}$. If one of them falls
upon the lower-right border, then the extremal points of $U(\gamma+1,\gamma)$
are the end of $I_{\xi}$ on the upper-left border and one of the extremal
points of $U(\gamma,\gamma)$, and the inclination of $U(\gamma+1,\gamma)$ is
between the inclination of $U(\gamma+1,\gamma)$ and the inclination of the
segment, which means it is between $\alpha$ and $\frac{\pi}{2}-\alpha$. If
both ends fall upon the lower-right border, the extremal points of
$U(\gamma+1,\gamma)$ are the extremal points of $U(\gamma,\gamma)$, which
means $U(\gamma+1,\gamma)$ has inclination $\alpha^{\prime\prime}$. Thus, the
inclination of $U(\gamma,\gamma)$ is between $\alpha$ and
$\frac{\pi}{2}-\alpha$.

If $U(\gamma+1,\gamma)\subset{\tilde{S}}_{\delta}$, the induction thesis is
satisfied. Thus we may assume $U(\gamma+1,\gamma)$ sticks outside
${\tilde{S}}_{\delta}$. Note that as $\theta_{n-2}(p)$,
$p\in{\mathbb{R}}_{x}\times{\mathbb{R}}_{y}$, is a coordinate-wise increasing
function from property (T4), one of the extremal points of
$U(\gamma+1,\gamma)$ has to lie outside ${\tilde{S}}_{\delta}$, and at least
one point of ${\tilde{S}}$ lies on the extremal line of $U(\gamma+1,\gamma)$.
Thus, the length of the segment of the extremal line contained in
$\tilde{K}_{+}$ is at least $\delta$.

Thus $U(\gamma+1,\gamma)$ has relevant length at least $\delta$, width at most
$w$ and inclination between $\alpha$ and $\frac{\pi}{2}-\alpha$. Thus from
Lemma 5.12 we know that $U(\gamma+1,\gamma)$ is ${\varepsilon}$-appropriate,
which means we completed the induction step.

###  6.3 The induction step for limit ordinals

For limit ordinals $\gamma$ the set $U(\gamma,\gamma+1)=U(0,1)$, the sets
$U(\gamma,\beta)$ for $\beta<\gamma$ are defined by
$U(\gamma,\beta)=U(\beta+1,\beta)$, and from the inductive assumption the
conditions for $U(\gamma,\beta)$ are met. We define $U(\gamma,\gamma)$ as the
intersection $\bigcap_{\beta<\gamma}U(\beta,\beta)$.

We have to check that $U(\gamma,\gamma)$ thus defined satisfies the induction
thesis. If any of the sets $U(\gamma^{\prime},\beta),\beta<\gamma^{\prime}$
was empty, then from the inductive assumption
$U(\gamma^{\prime}+1,\gamma^{\prime}+1)$ has $\mu_{2}$ measure 0 and thus
$U(\gamma,\gamma)$ has $\mu_{2}$ measure 0, which satisfies the conditions. If
$U(\beta,\beta)$ was not a strict lens set for some $\beta<\gamma$, then
$U(\gamma,\gamma)$ has measure 0, again satisfying the conditions. The case to
worry about is when $U(\gamma,\gamma)$ is a intersection of a descending
family of strict lens sets satisfying condition (5.1.2) and has a positive
$\mu_{2}$ measure.

A descending intersection of lens sets is a lens set — the circumscribed
rectangle is the intersection of circumscribed rectangles, the extremal points
belong to the intersection, and the intersection is convex. A descending
intersection of sets satisfying (5.1.2) with positive $\mu_{2}$ measure
satisfies (5.1.2) by property (T5). We have to prove that the intersection is
either a strict lens set, or appropriate. As $U(\gamma,\gamma)\subset U(0,0)$,
it is separated from $x_{-},x_{+},y_{-}$ and $y_{+}$. Thus we only have to
check it does not have a horizontal or vertical edge.

Suppose $U(\gamma,\gamma)$ has a horizontal or vertical edge $I$. We may
assume, without loss of generality, that $I$ is a horizontal edge. We will
assume it is an upper horizontal edge. In the case of the lower one, the proof
goes very similarily: every construction of new points is done centrally-
symetric, and every inequality is opposite. In one place, where the proof
significantly changes, we will say it explicitly.

Let $(x_{0},y_{0})$ be the left end of $I$ and $(x_{1},y_{0})$ the right end.
First we shall prove the following Lemma:

######  Lemma 6.2.

With the notation as previously we have ${\rm cl}I\cap{\rm