# Density estimate from below
in relation to a conjecture of A. Zygmund
on Lipschitz differentiation

Thierry De Pauw  School of Mathematical Sciences
Shanghai Key Laboratory of PMMP
East China Normal University
500 Dongchuang Road
Shanghai 200062
P.R. of China
and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
3663 Zhongshan Road North
Shanghai 200062
China Université Paris Diderot
Sorbonne Université
CNRS
Institut de Mathématiques de Jussieu – Paris Rive Gauche, IMJ-PRG
F-75013, Paris
France thdepauw<EMAIL_ADDRESS>
###### Abstract.

Letting $A\subseteq\mathbb{R}^{n}$ be Borel measurable and
$\mathbf{W}_{0}:A\to\mathbb{G}(n,m)$ Lipschitzian, we establish that

$\limsup_{r\to
0^{+}}\frac{\mathscr{H}^{m}\left[A\cap\mathbf{B}(x,r)\cap(x+\mathbf{W}_{0}(x))\right]}{\boldsymbol{\alpha}(m)r^{m}}\geqslant\frac{1}{2^{n}},$

for $\mathscr{L}^{n}$-almost every $x\in A$. In particular, it follows that
$A$ is $\mathscr{L}^{n}$-negligible if and only if
$\mathscr{H}^{m}(A\cap(x+\mathbf{W}_{0}(x))=0$, for $\mathscr{L}^{n}$-almost
every $x\in A$.

###### Key words and phrases:

Lebesgue measure, Nikodým set, Negligible set, Derivation basis, Zygmund
conjecture, Lipschitz differentiation

###### 2010 Mathematics Subject Classification:

Primary 28A75,26B15

The author was partially supported by the Science and Technology Commission of
Shanghai (No. 18dz2271000).

###### Contents

  1. 1 Foreword
  2. 2 Sketch of proof
  3. 3 Preliminaries
  4. 4 Common setting
  5. 5 Two fibrations
  6. 6 Upper bound for $\mathscr{Y}_{E}\mathbf{W}$ and $\mathscr{Z}_{E}\mathbf{W}$
  7. 7 Lower bound for $\mathscr{Y}_{E}\mathbf{W}$ and $\mathscr{Z}_{E}\mathbf{W}$
  8. 8 Proof of the theorems

##  1\. Foreword

Let $\mathscr{S}_{x}$ be the set of squares centered at $x\in\mathbb{R}^{2}$.
The Lebesgue density theorem states that if $f:\mathbb{R}^{2}\to\mathbb{R}$ is
Lebesgue summable, then

$f(x)=\lim_{\begin{subarray}{c}S\in\mathscr{S}_{x}\\\
\operatorname{\mathrm{diam}}S\to
0^{+}\end{subarray}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S}fd\mathscr{L}^{2},$
($*$)

for $\mathscr{L}^{2}$-almost every $x\in\mathbb{R}^{2}$. This consequence of
the Vitali covering theorem fails if $\mathscr{S}_{x}$ is replaced with
$\mathscr{R}_{x}$, the set of rectangles centered at $x$ of arbitrary
direction and eccentricity. It fails even for some indicator function
$f=\mathbbm{1}_{A}$. Indeed, Nikodým [12] defined a set
$A\subseteq[0,1]\times[0,1]$ of full measure with the following property: For
every $x\in A$, there exists a line $L(x)$ such that $A\cap L(x)=\\{x\\}$. In
fact, Nikodým’s example shows that the Lebesgue density theorem fails if we
replace $\mathscr{S}_{x}$ with $\mathscr{R}_{x}^{L}$, the set of rectangles of
arbitrary eccentricity, centered at $x$, and one side of which is parallel to
$L(x)$. Furthermore [4, Chap. IV Theorem 3.5], replacing $A$ by a
nonnegligible subset of $A$, one may choose $L$ to be continuous with respect
to $x$. Yet, if $L$ is constant, then the Lebesgue density theorem with
respect to $\mathscr{R}_{x}^{L}$ holds, for $p$-summable functions $f$,
$1<p\leqslant\infty$, by virtue of a theorem of Zygmund, though the
corresponding version of the Vitali covering theorem fails, according to an
example of H. Bohr [4, Chap. IV Theorem 1.1]. This raises the question: What
regularity condition of $x\mapsto L(x)$ guarantees that the Lebesgue density
theorem holds with respect to $\mathscr{R}_{x}^{L}$ for, say, functions that
are square summable? The following version is a conjecture reportedly [11]
attributed to Zygmund. Assume that $x\mapsto L(x)$ is a Lipschitzian field of
lines, with $x\in L(x)$, and $f:\mathbb{R}^{2}\to\mathbb{R}$ is square
summable. Is it true that

$f(x)=\lim_{r\to
0^{+}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{L(x)\cap\mathbf{B}(x,r)}fd\mathscr{H}^{1},$

for $\mathscr{L}^{2}$-almost every $x\in\mathbb{R}^{2}$? Here,
$\mathscr{H}^{1}$ is the 1-dimensional Hausdorff measure.

E.M. Stein raised the singular integral variant of this conjecture. Both have
received much attention from the harmonic analysis community. To the author’s
knowledge, most results recorded so far in the literature, via the maximal
function approach, assume some extra regularity property of $L$ – namely, that
$L$ be $C^{1}$ together with a hypothesis on the variation of the derivative –
see, for instance, [1] and [11].

In this paper we offer a novel approach – a kind of change of variable based
on an appropriate fibration and the coarea formula. This allows for treating
the case when $L$ is Lipschitzian, without further restriction. We obtain the
following lower density bound for indicator functions. Let $x\mapsto L(x)$ be
a Lipschitzian field of lines, with $x\in L(x)$, and let
$A\subseteq\mathbb{R}^{2}$ be Lebesgue measurable. Then,

$\limsup_{r\to 0^{+}}\frac{\mathscr{H}^{1}(A\cap B(x,r)\cap
L(x))}{2r}\geqslant\frac{1}{4},$

for $\mathscr{L}^{2}$-almost every $x\in A$.  Incidentally, the following
corollary – a nonparallel version of Fubini theorem – seems to be new as well.
The set $A$ is Lebesgue negligible if and only if $\mathscr{H}^{1}(A\cap
L(x))=0$, for $\mathscr{L}^{2}$-almost every $x\in\mathbb{R}^{2}$.

Our results hold in any dimension and codimension.

##  2\. Sketch of proof

Let $A$ be a subset of Euclidean space $\mathbb{R}^{n}$, $n\geqslant 2$, and
let $\mathscr{L}^{n}$ be the Lebesgue outer measure. We start by considering
the following weak question: Can one tell whether $A$ is Lebesgue negligible
from the knowledge only of its trace on each member of some given collection
of “lower dimensional” subsets $\Gamma_{i}\subseteq\mathbb{R}^{n}$, $i\in I$.
One expects that if $A\cap\Gamma_{i}$ is “negligible in the dimension of
$\Gamma_{i}$”, for each $i\in I$, then $\mathscr{L}^{n}(A)=0$. Of course, a
necessary condition is that the sets $\Gamma_{i}$ cover almost all of $A$,
i.e., $\mathscr{L}^{n}(A\setminus\cup_{i\in I}\Gamma_{i})=0$. Consider, for
instance, $n=2$, $I=\mathbb{R}$, and $\Gamma_{t}=\\{t\\}\times\mathbb{R}$, for
$t\in\mathbb{R}$, the collection of all vertical lines in the plane. It is not
true in general that if $A\subseteq\mathbb{R}^{2}$ and $A\cap\Gamma_{t}$ is a
singleton, for each $t\in\mathbb{R}$, then $\mathscr{L}^{2}(A)=0$. There
exist, indeed, functions $f:\mathbb{R}\to\mathbb{R}$ whose graph $A$ satisfies
$\mathscr{L}^{2}(A)>0$ – see, e.g., [10, Chapter 2 Theorem 4] for an example
due to Sierpiński. In order to rule out such examples, we will henceforth
assume that $A\subseteq\mathbb{R}^{n}$ is Borel measurable. In that case, the
theorem of Fubini, together with the invariance of the Lebesgue measure under
orthogonal transformations, imply the following. Given an integer $1\leqslant
m\leqslant n-1$, if $(\Gamma_{i})_{i\in I}$ is the collection of all
$m$-dimensional affine subspaces of $\mathbb{R}^{n}$ of some fixed direction,
and if $\mathscr{H}^{m}(A\cap\Gamma_{i})=0$ for all $i\in I$, then
$\mathscr{L}^{n}(A)=0$. Here, $\mathscr{H}^{m}$ denotes the $m$-dimensional
Hausdorff measure. A special feature of this collection $(\Gamma_{i})_{i\in
I}$ is that it partitions $\mathbb{R}^{n}$, its members being the level sets
$f^{-1}\\{y\\}$, $y\in\mathbb{R}^{n-m}$, of a “nice map”
$f:\mathbb{R}^{n}\to\mathbb{R}^{n-m}$, indeed, an orthogonal projection. This
is an occurrence of the following more general situation when $f$ and its
leaves $f^{-1}\\{y\\}$ are allowed to be nonlinear. The coarea formula due to
Federer [8] asserts that if $f:\mathbb{R}^{n}\to\mathbb{R}^{n-m}$ is
Lipschitzian and $A\subseteq\mathbb{R}^{n}$ is Borel measurable, then

$\int_{A}Jf(x)d\mathscr{L}^{n}(x)=\int_{\mathbb{R}^{n-m}}\mathscr{H}^{m}\left(A\cap
f^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y).$

Thus, if the Jacobian coarea factor $Jf$ is positive, $\mathscr{L}^{n}$-almost
everywhere in $A$, then the collection
$\left(f^{-1}\\{y\\}\right)_{y\in\mathbb{R}^{n-m}}$ is suitable for detecting
whether or not $A$ is Lebesgue null. At $\mathscr{L}^{n}$-almost all
$x\in\mathbb{R}^{n}$, the map $f$ is differentiable, by virtue of Rademacher’s
theorem, and

$Jf(x)=\sqrt{\left|\det\left(Df(x)\circ
Df(x)^{*}\right)\right|}=\left\|\wedge_{n-m}Df(x)\right\|,$

see [6, Chapter 3 §4] and [9, 3.2.1 and 3.2.11].

In this paper, we focus on the case when $\Gamma_{i}$, $i\in I$, are affine
subspaces of $\mathbb{R}^{n}$, but not necessarily members of a partition of
the ambient space. Specifically, we assume that with each $x\in\mathbb{R}^{n}$
is associated an $m$-dimensional affine subspace $\mathbf{W}(x)$ of
$\mathbb{R}^{n}$ containing $x$. Given a Borel set $A\subseteq\mathbb{R}^{n}$,
the question whether

$\textit{If }\mathscr{H}^{m}(A\cap\mathbf{W}(x))=0\text{, for all }x\in
A\textit{, then }\mathscr{L}^{n}(A)=0\,,$ (1)

has a negative answer, in view of Nikodým’s set $A\subseteq\mathbb{R}^{2}$
evoked in the previous section. Indeed, corresponding to this set $A$, there
exists a field of lines $x\mapsto\mathbf{W}(x)$ such that
$A\cap\mathbf{W}(x)=\\{x\\}$, for all $x\in A$. In this context, a selection
theorem due to von Neumann implies that (possibly considering a smaller, non
Lebesgue null, Borel subset of $A$) the correspondence $x\mapsto\mathbf{W}(x)$
can be chosen to be Borel measurable (see 3.19) and, in turn, it can be chosen
to be continuous, according to Lusin’s theorem. Nonetheless, when $\mathbf{W}$
is Lipschitzian, the situation improves, as illustrated in our theorem below;
$\mathbb{G}(n,m)$ is the Grassmannian manifold consisting of $m$-dimensional
vector subspaces of $\mathbb{R}^{n}$.

######  Theorem. —

Assume $\mathbf{W}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$ is Lipschitzian and
$A\subseteq\mathbb{R}^{n}$ is Borel measurable. The following are equivalent.

  1. (A)

$\mathscr{L}^{n}(A)=0$.

  2. (B)

For $\mathscr{L}^{n}$ almost every $x\in A$,
$\mathscr{H}^{m}\left(A\cap(x+\mathbf{W}_{0}(x))\right)=0$.

  3. (C)

For $\mathscr{L}^{n}$ almost every $x\in\mathbb{R}^{n}$,
$\mathscr{H}^{m}\left(A\cap(x+\mathbf{W}_{0}(x))\right)=0$.

As should be apparent from the discussion above, one difficulty stands with
the fact that the affine $m$-planes $\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$ may
not be disjointed. Nevertheless, they locally are, in the following sense.
Given $x_{0}\in A$ there exist a neighborhood $U$ of $x_{0}$ and Lipschitzian
maps $\mathbf{w}_{1},\ldots,\mathbf{w}_{m}:U\to\mathbb{R}^{n}$ so that
$\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x)$ is an orthonomal frame spanning
$\mathbf{W}_{0}(x)$, for $x\in U$, and the map $\Phi:(V_{x_{0}}\cap
U)\times\mathbb{R}^{m}\to\mathbb{R}^{n}:(\xi,t)\mapsto\xi+\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(\xi)$
is a lipeomorphism of a neighborhood of $x_{0}$ onto its image – here,
$V_{x_{0}}=x_{0}+V$ and $V\in\mathbb{G}(n,n-m)$ is close to
$\mathbf{W}_{0}(x_{0})^{\perp}$. This, and applications of Fubini’s theorem,
yield $(B)\Rightarrow(A)$ in the theorem above (see 8.2).

However, we aim at obtaining a quantative version of this result, that the
change of variable just described does not seem to provide. A natural route is
to reduce the problem to applying the coarea formula, by spreading out the
$\mathbf{W}(x)$’s in a disjointed way, in a higher dimensional space – i.e.,
adding a variable $u\in\mathbf{W}(x)$ to the given $x\in\mathbb{R}^{n}$ and
considering $\mathbf{W}(x)$ as a fiber above the base space $\mathbb{R}^{n}$.
We thus define

$\Sigma=\mathbb{R}^{n}\times\mathbb{R}^{n}\cap\\{(x,u):x\in E\text{ and
}u\in\mathbf{W}(x)\\},$

where $E\subseteq\mathbb{R}^{n}$ is Borel measurable. This set is
$(n+m)$-rectifiable, owing to the Lipschitz continuity of $\mathbf{W}$. It is
convenient to assume that $\mathscr{L}^{n}(E)<\infty$, so that

$\phi_{E}(B)=\int_{E}\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)d\mathscr{L}^{n}(x),$
(2)

$B\subseteq\mathbb{R}^{n}$, is a locally finite Borel measure 3.16. Now,
$\Sigma$ was precisely set up so that, for each $x\in E$,

$\mathscr{H}^{m}\left(\Sigma\cap\pi_{2}^{-1}(B)\cap\pi_{1}^{-1}\\{x\\}\right)=\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right),$

where $\pi_{1}$ and $\pi_{2}$ denote the projections of
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ to the $x$ and $u$ variable,
respectively. Abbreviating $\Sigma_{B}=\Sigma\cap\pi_{2}^{-1}(B)$, the coarea
formula yields

$\phi_{E}(B)=\int_{\Sigma_{B}}J_{\Sigma}\pi_{1}d\mathscr{H}^{n+m}.$ (3)

A simple calculation 5.4 shows that $J_{\Sigma}\pi_{2}>0$,
$\mathscr{H}^{n+m}$-almost everywhere on $\Sigma_{B}$. Since also

$\int_{\Sigma_{B}}J_{\Sigma}\pi_{2}d\mathscr{H}^{n+m}=\int_{B}\mathscr{H}^{m}\left(\Sigma_{B}\cap\pi_{2}^{-1}\\{u\\}\right)d\mathscr{L}^{n}(u),$
(4)

the implication $(A)\Rightarrow(C)$ above should now be clear: Letting $B=A$
and $E=\mathbf{B}(0,R)$, one infers from hypothesis (A) and (4) that
$\mathscr{H}^{n+m}(\Sigma_{B})=0$, whence, $\phi_{\mathbf{B}(0,R)}(A)=0$, by
(3), and, in turn, conclusion (C) ensues from (2).

In order to establish that $(B)\Rightarrow(A)$ by using this new change of
variable, we need to observe 5.2 that $J_{\Sigma}\pi_{1}>0$,
$\mathscr{H}^{n+m}$-almost everywhere, and, ideally, to show that
$\mathscr{H}^{m}\left(\Sigma_{B}\cap\pi_{2}^{-1}\\{u\\}\right)>0$, for almost
every $u\in B$. This last part involves some difficulty. To understand this,
we let $m=n-1$, in order to keep the notations short. Now $u\in\mathbf{W}(x)$
iff $u-x\in\mathbf{W}_{0}(x)$ iff $\langle\mathbf{v}_{0}(x),x-u\rangle=0$,
where $\mathbf{v}_{0}(x)\in\mathbf{W}_{0}(x)^{\perp}$ is, say, a unit vector.
Abbreviating $g_{u}(x)=\langle\mathbf{v}_{0}(x),x-u\rangle$, we infer that

$\mathscr{H}^{m}\left(\Sigma_{B}\cap\pi_{2}^{-1}\\{u\\}\right)=\mathscr{H}^{m}\left(E\cap
g_{u}^{-1}\\{0\\}\right).$

The problem remains that two of the nonlinear $m$ sets $E\cap
g_{u}^{-1}\\{0\\}$ and $E\cap g_{u^{\prime}}^{-1}\\{0\\}$ may intersect,
thereby preventing another application of the coarea formula when looking out
for their lower bound. Yet, we already know that

$\phi_{E}(B)=\int_{B}\mathscr{Z}_{E}\mathbf{W}d\mathscr{L}^{n},$

where $\mathscr{Z}_{E}\mathbf{W}$ is a Radon-Nikodým derivative (see 5.6) and
also that $(\mathscr{Z}_{E}\mathbf{W})(u)$ is comparable to
$\mathscr{H}^{m}\left(E\cap g_{u}^{-1}\\{0\\}\right)$ (see 5.8). Adding an
extra variable $y$ to the fibered space $\Sigma$, 5.10, we improve on this by
showing that

$(\mathscr{Z}_{E}\mathbf{W})(u)\geqslant\boldsymbol{\eta}(n,m)\liminf_{j}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{-j^{-1}}^{j^{-1}}\mathscr{H}^{m}\left(E\cap
g_{u}^{-1}\\{y\\}\right)d\mathscr{L}^{1}(y)=\boldsymbol{\eta}(n,m)(\mathscr{Y}_{E}\mathbf{W})(u),$

where the last equality defines $\mathscr{Y}_{E}\mathbf{W}$, and
$\boldsymbol{\eta}(n,m)>0$ is a lower bound for the coarea Jacobian factor of
the restriction to the fibered space of the projection $(x,u,y)\mapsto(x,y)$
(see 5.12 and 5.16). We are reduced to showing that
$\mathscr{Y}_{E}\mathbf{W}>0$, almost everywhere. The reason why this holds is
the following. Fix a Borel set $Z\subseteq\mathbb{R}^{n}$,
$x_{0}\in\mathbb{R}^{n}$ and $r>0$. Let $\mathbf{C}_{\mathbf{W}}(x_{0},r)$ be
the cylindrical box consisting of those $x\in\mathbb{R}^{n}$ such that
$\left|P_{\mathbf{W}_{0}(x_{0})}(x-x_{0})\right|\leqslant r$ and
$\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(x-x_{0})\right|\leqslant r$. We want
to find a lower bound for

$\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\left(\mathscr{Y}_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\right)(u)d\mathscr{L}^{n}(u).$

To this end, we fix $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$ and we let
$V_{z}=\mathbb{R}^{n}\cap\\{x_{0}+z+s\mathbf{v}_{0}(x_{0}):-r\leqslant
s\leqslant r\\}$ denote the corresponding vertical line segment. According to
Fubini’s theorem we are reduced to estimating

$\int_{V_{z}}\left(\mathscr{Y}_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\right)(u)d\mathscr{H}^{1}(u).$

By virtue of Vitali’s covering theorem, we can find a disjointed family of
line segments $I_{1},I_{2},\ldots$, covering almost all $V_{z}$, such that the
above integral nearly equals

$\sum_{k}\mathscr{H}^{1}(I_{k})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{I_{k}}\mathscr{H}^{m}\left(Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap
g_{u_{k}}^{-1}\\{y\\}\right)d\mathscr{L}^{1}(y)\\\
\cong\sum_{k}\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap
g_{u_{k}}^{-1}(I_{k})}\left|\nabla
g_{u_{k}}(x)\right|d\mathscr{L}^{n}(x)\cong\mathscr{L}^{n}\left(Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right),$

where the first near equality follows from the coarea formula, and the second
one because $\left|\nabla g_{u_{k}}\right|\cong 1$ at small scales (see 3.12)
and the “nonlinear horizontal stripes” $g_{u_{k}}^{-1}(I_{k})$ are nearly
pairwise disjoint. Verification of these claims takes up sections 6 and 7.
Now, we reach a contradiction if
$Z=\mathbb{R}^{n}\cap\\{\mathscr{Y}_{E}\mathbf{W}=0\\}$ is assumed to have
$\mathscr{L}^{n}(Z)>0$, and $x_{0}$ is a point of density of $Z$.

Along this path, we have, in fact, achieved a quantitative version of these
observations. Indeed, assuming that $\limsup_{r\to
0^{+}}\frac{\mathscr{H}^{m}(A\cap\mathbf{B}(x,r)\cap\mathbf{W}(x))}{\boldsymbol{\alpha}(m)r^{m}}<\frac{\boldsymbol{\eta}(n,m)}{2^{m}}$,
for $\mathscr{L}^{n}$ positively many $x\in A$, we start by choosing
$E\subseteq A$ with the same property, whose diameter is small enough for the
various estimates above to hold, and we refer to Egoroff’s theorem to select
$\varepsilon>0$ so that the set

$Z=E\cap\left\\{\limsup_{r\to
0^{+}}\frac{\mathscr{H}^{m}(A\cap\mathbf{B}(x,r)\cap\mathbf{W}(x))}{\boldsymbol{\alpha}(m)r^{m}}<(1-\varepsilon)\frac{\boldsymbol{\eta}(n,m)}{(2+\varepsilon)^{m}}\right\\}$

has positive $\mathscr{L}^{n}$ measure. Thus, for
$0<\hat{\varepsilon}\ll\varepsilon$, there exists $x_{0}\in Z$ and there
exists $r>0$ as small as we please so that

$\left(1-\hat{\varepsilon}\right)\mathscr{L}^{m}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\leqslant\mathscr{L}^{n}\left(Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\lesssim\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\frac{\mathscr{Y}_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)}{\boldsymbol{\alpha}(m)r^{m}}d\mathscr{L}^{n}(u)\\\
\lesssim\frac{1}{\boldsymbol{\eta}(n,m)}\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\frac{\mathscr{H}^{m}(Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap\mathbf{W}(x))}{\boldsymbol{\alpha}(m)r^{m}}d\mathscr{L}^{n}(x)\\\
\leqslant\frac{1}{\boldsymbol{\eta}(n,m)}\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\frac{\mathscr{H}^{m}(Z\cap\mathbf{B}(x,2r+o(r))\cap\mathbf{W}(x))}{\boldsymbol{\alpha}(m)r^{m}}d\mathscr{L}^{n}(x)\\\
\lesssim\frac{1}{\boldsymbol{\eta}(n,m)}\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}(1-\varepsilon)\frac{\boldsymbol{\eta}(n,m)}{(2+\varepsilon)^{m}}d\mathscr{L}^{n}(x)\leqslant(1-C\varepsilon)\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right),$

where the symbol $\lesssim$ means that we loose a multiplicative factor
$\lambda>1$ as close as we wish to 1, according to the scale $r$. The above
contradiction proves our main result 8.4, namely,

######  Theorem. —

Assume $\mathbf{W}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$ is Lipschitzian and
$A\subseteq\mathbb{R}^{n}$ is Borel measurable. Abbreviating
$\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$, it follows that

$\limsup_{r\to
0^{+}}\frac{\mathscr{H}^{m}\big{(}A\cap\mathbf{B}(x,r)\cap\mathbf{W}(x)\big{)}}{\boldsymbol{\alpha}(m)r^{m}}\geqslant\frac{1}{2^{n}},$

for $\mathscr{L}^{n}$-almost every $x\in A$.

I extend my warm thanks to Jean-Christophe Léger for carefully reading an
early version of the manuscript.

##  3\. Preliminaries

######  3.1. —

In this paper, $1\leqslant m\leqslant n-1$ are integers. The ambient space is
$\mathbb{R}^{n}$. The canonical inner product of
$x,x^{\prime}\in\mathbb{R}^{n}$ is denoted $\langle x,x^{\prime}\rangle$ and
the corresponding Euclidean norm of $x$ is $|x|$. If
$S\subseteq\mathbb{R}^{n}$, we let $\mathscr{B}(S)$ be the $\sigma$-algebra of
Borel measurable subsets of $S$.

######  3.2Hausdorff measure. —

We let $\mathscr{L}^{n}$ be the Lebesgue outer measure in $\mathbb{R}^{n}$ and
$\boldsymbol{\alpha}(n)=\mathscr{L}^{n}(\mathbf{B}(0,1))$. For
$S\subseteq\mathbb{R}^{n}$, we abbreviate
$\zeta^{m}(S)=\boldsymbol{\alpha}(m)2^{-m}(\operatorname{\mathrm{diam}}S)^{m}$.
Given $0<\delta\leqslant\infty$, we call $\delta$-cover of
$A\subseteq\mathbb{R}^{n}$ a finite or countably infinite family
$(S_{j})_{j\in J}$ of subsets of $\mathbb{R}^{n}$ such that
$A\subseteq\cup_{j\in J}S_{j}$ and
$\operatorname{\mathrm{diam}}S_{j}\leqslant\delta$, for every $j\in J$. We
define

$\mathscr{H}^{m}_{\delta}(A)=\inf\left\\{\sum_{j\in
J}\zeta^{m}(S_{j}):(S_{j})_{j\in J}\text{ is a $\delta$-cover of $A$
}\right\\}$

and $\mathscr{H}^{m}(A)=\lim_{\delta\to
0^{+}}\mathscr{H}^{m}_{\delta}(A)=\sup_{\delta>0}\mathscr{H}^{m}_{\delta}(A)$.
Thus, $\mathscr{H}^{m}$ is the $m$-dimensional Hausdorff outer measure in
$\mathbb{R}^{n}$.

  1. (1)

If $(K_{k})_{k}$ is a sequence of nonempty compact subsets of $\mathbb{R}^{n}$
converging in Hausdorff distance to $K$, then
$\mathscr{H}^{m}_{\infty}(K)\geqslant\limsup_{k}\mathscr{H}^{m}_{\infty}(K_{k})$.

Given $\varepsilon>0$, choose an $\infty$-cover $(S_{j})_{j\in\mathbb{N}}$ of
$K$ such that
$\mathscr{H}^{m}_{\infty}(K)+\varepsilon\geqslant\sum_{j}\zeta^{m}(S_{j})$.
Since $\lim_{r\to 0^{+}}\zeta^{m}(\mathbf{U}(S_{j},r))=\zeta^{m}(S_{j})$, for
each $j\in\mathbb{N}$, we can choose an open set $U_{j}$ containing $S_{j}$
such that $\zeta^{m}(U_{j})\leqslant\varepsilon 2^{-j}+\zeta^{m}(S_{j})$.
Since $U=\cup_{j}U_{j}$ is open, there exists a positive integer $k_{0}$ such
that $K_{k}\subseteq U$, whenever $k\geqslant k_{0}$. Thus, in that case
$(U_{j})_{j}$ is an $\infty$-cover of $K_{k}$ and, therefore,
$\mathscr{H}^{m}_{\infty}(K_{k})\leqslant\sum_{j}\zeta^{m}(U_{j})\leqslant
2\varepsilon+\mathscr{H}^{m}_{\infty}(K)$. Take the limit superior of the left
hand side, as $k\to\infty$, and then let $\varepsilon\to 0$.

  1. (2)

For all $A\subseteq\mathbb{R}^{m}$, one has
$\mathscr{L}^{m}(A)=\mathscr{H}^{m}(A)=\mathscr{H}^{m}_{\infty}(A)$.

It suffices to note that
$\mathscr{H}^{m}(A)\geqslant\mathscr{H}^{m}_{\infty}(A)\geqslant\mathscr{L}^{m}(A)\geqslant\mathscr{H}^{m}(A)$.
The first inequality is trivial; the second one follows from the isodiametric
inequality [9, 2.10.33]; the last one is a consequence of the Vitali covering
theorem [6, Chapter 2 §2 Theorem 2].

  1. (3)

If $W\subseteq\mathbb{R}^{n}$ is an $m$-dimensional affine subspace and
$A\subseteq W$, then $\mathscr{H}^{m}(A)=\mathscr{H}^{m}_{\infty}(A)$.

Let $\mathfrak{H}^{m}$ denote the $m$-dimensional Hausdorff outer measure in
the metric space $W$. In other words,

$\mathfrak{H}^{m}_{\delta}(A)=\inf\left\\{\sum_{j\in
J}\zeta^{m}(S_{j}):(S_{j})_{j\in J}\text{ is a $\delta$- cover of $A$ and
$S_{j}\subseteq W$, for all $j\in J$}\right\\},$

and $\mathfrak{H}^{m}(A)=\sup_{\delta>0}\mathfrak{H}^{m}_{\delta}(A)$. It is
elementary to observe that $\mathfrak{H}^{m}(A)=\mathscr{H}^{m}(A)$ and that
$\mathfrak{H}^{m}_{\infty}(A)=\mathscr{H}^{m}_{\infty}(A)$. Now, if
$f:W\to\mathbb{R}^{m}$ is an isometry, then
$\mathscr{H}^{m}(A)=\mathfrak{H}^{m}(A)=\mathscr{H}^{m}(f(A))=\mathscr{H}^{m}_{\infty}(f(A))=\mathfrak{H}^{m}_{\infty}(A)=\mathscr{H}^{m}_{\infty}(A)$,
where the third equality follows from claim (2) above.

######  3.3Coarea formula. —

We recall two versions of the coarea formula. First, if
$A\subseteq\mathbb{R}^{n}$ is $\mathscr{L}^{n}$-measurable and
$f:A\to\mathbb{R}^{n-m}$ is Lipschitzian, then
$\mathbb{R}^{n-m}\to[0,\infty]:y\mapsto\mathscr{H}^{m}\left(A\cap
f^{-1}\\{y\\}\right)$ is $\mathscr{L}^{n-m}$\- measurable and

$\int_{A}Jf(x)d\mathscr{L}^{n}(x)=\int_{\mathbb{R}^{n-m}}\mathscr{H}^{m}\left(A\cap
f^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y).$

Here, the coarea Jacobian factor is well-defined $\mathscr{L}^{n}$-almost
everywhere, according to Rademacher’s theorem, and equals

$Jf(x)=\sqrt{\left|\det\left(Df(x)\circ
Df(x)^{*}\right)\right|}=\left\|\wedge_{n-m}Df(x)\right\|;$

see, for instance, [6, Chapter 3 §4].

Secondly, if $A\subseteq\mathbb{R}^{p}$ is $\mathscr{H}^{n}$-measurable and
countably $(\mathscr{H}^{n},n)$-rectifiable and if $f:A\to\mathbb{R}^{n-m}$ is
Lipschitzian, then
$\mathbb{R}^{n-m}\to[0,\infty]:y\mapsto\mathscr{H}^{m}\left(A\cap
f^{-1}\\{y\\}\right)$ is $\mathscr{L}^{n-m}$-measurable and

$\int_{A}J_{A}f(x)d\mathscr{H}^{n}(x)=\int_{\mathbb{R}^{n-m}}\mathscr{H}^{m}\left(A\cap
f^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y).$

In order to state a formula for the coarea Jacobian factor $J_{A}f(x)$ of $f$
relative to $A$, we consider a point $x\in A$, where $A$ admits an approximate
$n$-dimensional tangent space $T_{x}A$ and $f$ is differentiable at $x$ along
$A$. Letting $L:T_{x}A\to\mathbb{R}^{n-m}$ denote the derivative of $f$ at
$x$, we have

$J_{A}f(x)=\sqrt{\left|\det\left(L\circ
L^{*}\right)\right|}=\left\|\wedge_{n-m}L\right\|;$

see, for instance, [9, 3.2.22].

In both cases, it is useful to recall the following. If $L:V\to V^{\prime}$ is
a linear map between two inner product spaces $V$ and $V^{\prime}$, then

$\|\wedge_{k}L\|=\sup\left\\{\langle\wedge_{k}L,\xi\rangle:\xi\in\wedge_{k}V\text{
and }|\xi|=1\right\\}.$ (5)

On the one hand, $\|\wedge_{k}L\|\leqslant\|L\|^{k}$ [9, 1.7.6] and
$\|L\|\leqslant\operatorname{\mathrm{Lip}}f$, with $L$ as above. On the other
hand, if $v_{1},\ldots,v_{k}$ are linearly independent vectors of $V$, then

$\|\wedge_{k}L\|\geqslant\frac{|L(v_{1})\wedge\ldots\wedge
L(v_{k})|}{|v_{1}\wedge\ldots\wedge v_{k}|}.$ (6)

Finally, we observe that both coarea formulæ hold true when $f$ is merely
locally Lipschitzian, according to the monotone convergence theorem.

######  3.4Grassmannian. —

We let $\mathbb{G}(n,m)$ be the set whose members are the $m$-dimensional
vector subspaces of $\mathbb{R}^{n}$. With $W\in\mathbb{G}(n,m)$, we associate
$P_{W}:\mathbb{R}^{n}\to\mathbb{R}^{n}$, the orthogonal projection on $W$. We
give $\mathbb{G}(n,m)$ the structure of a compact metric space by letting
$d(W_{1},W_{2})=\|P_{W_{1}}-P_{W_{2}}\|$, where $\|\cdot\|$ is the operator
norm. If $W\in\mathbb{G}(n,m)$, then $W^{\perp}\in\mathbb{G}(n,n-m)$ satisfies
$P_{W}+P_{W^{\perp}}=\operatorname{\mathrm{id}}_{\mathbb{R}^{n}}$, therefore,
$\mathbb{G}(n,m)\to\mathbb{G}(n,n-m):W\mapsto W^{\perp}$ is an isometry. The
bijective correspondence
$\varphi:\mathbb{G}(n,m)\to\operatorname{\mathrm{Hom}}(\mathbb{R}^{n},\mathbb{R}^{n}):W\mapsto
P_{W}$ identifies $\mathbb{G}(n,m)$ with the submanifold
$M_{n,m}=\operatorname{\mathrm{Hom}}(\mathbb{R}^{n},\mathbb{R}^{n})\cap\\{L:L\circ
L=L\,,\,L^{*}=L\text{, and }\operatorname{\mathrm{trace}}L=m\\}$. There exists
an open neighborhood $V$ of $M_{n,m}$ in
$\operatorname{\mathrm{Hom}}(\mathbb{R}^{n},\mathbb{R}^{n})$ and a
Lipschitzian retraction $\rho:V\to M_{n,m}$, according, for instance, to [9,
3.1.20]. Therefore, if $S\subseteq\mathbb{R}^{n}$ and
$\mathbf{W}_{0}:S\to\mathbb{G}(n,m)$ is Lipschitzian, then there exist an open
neighborhood $U$ of $S$ in $\mathbb{R}^{n}$ and a Lipschitzian extension
$\widehat{\mathbf{W}}_{0}:U\to\mathbb{G}(n,m)$ of $\mathbf{W}_{0}$. Indeed,
$\varphi\circ\mathbf{W}_{0}$ admits a Lipschitzian extension
$Y:\mathbb{R}^{n}\to\operatorname{\mathrm{Hom}}(\mathbb{R}^{n},\mathbb{R}^{n})$
– see, e.g., [9, 2.10.43] – and it suffices to let $U=Y^{-1}(V)$ and
$\widehat{\mathbf{W}}_{0}=\rho\circ\left(Y|_{U}\right)$.

######  3.5Orthonormal frames. —

We let $\mathbb{V}(n,m)$ be the set of orthonormal $m$-frames in
$\mathbb{R}^{n}$, i.e.,
$\mathbb{V}(n,m)=(\mathbb{R}^{n})^{m}\cap\\{(w_{1},\ldots,w_{m}):\text{ the
family }w_{1},\ldots,w_{m}\text{ is orthonormal}\\}$. We will consider it as a
metric space, with its structure inherited from $(\mathbb{R}^{n})^{m}$.

######  3.6. —

Let $\mathscr{V}\subseteq\mathbb{G}(n,m)$ be a nonempty closed set such that
$\operatorname{\mathrm{diam}}\mathscr{V}<1$. There exists a Lipschitzian map
$\Xi:\mathscr{V}\to\mathbb{V}(n,m)$ such that
$W=\operatorname{\mathrm{span}}\\{\Xi_{1}(W),\ldots,\Xi_{m}(W)\\}$, for every
$W\in\mathscr{V}$.

###### Proof.

Pick arbitrarily $W_{0}\in\mathscr{V}$. If $W\in\mathscr{V}$, then the map
$W_{0}\to W:w\mapsto P_{W}(w)$ is bijective: If $w\in W_{0}\setminus\\{0\\}$,
then $\left|P_{W}(w)-w\right|=\left|P_{W}(w)-P_{W_{0}}(w)\right|<|w|$, thus,
$P_{W}(w)\neq 0$. Letting $w_{1},\ldots,w_{m}$ be an arbitrary basis of
$W_{0}$, it follows that, for each $W\in\mathscr{V}$, the vectors
$w_{i}(W)=P_{W}(w_{i})$, $i=1,\ldots,m$, constitute a basis of $W$.
Furthermore, the maps $w_{i}:\mathscr{V}\to\mathbb{R}^{n}$ are Lipschitzian:
$\left|w_{i}(W)-w_{i}(W^{\prime})\right|=\left|P_{W}(w_{i})-P_{W^{\prime}}(w_{i})\right|\leqslant
d(W,W^{\prime})|w_{i}|$. We apply the Gram-Schmidt process:

$\overline{w}_{1}(W)=w_{1}(W)\quad\text{ and
}\quad\overline{w}_{i}(W)=w_{i}(W)-\sum_{j=1}^{i-1}\langle
w_{i}(W),\overline{w}_{j}(W)\rangle\overline{w}_{j}(W)\,,\,\,i=2,\ldots,m,$

so that $\overline{w}_{1}(W),\ldots,\overline{w}_{m}(W)$ is readily an
orthogonal basis of $W$ depending upon $W$ in a Lipschitzian way. Since each
$\left|\overline{w}_{i}\right|$ is bounded away from zero on $\mathscr{V}$,
the formula
$\Xi_{i}(W)=\left|\overline{w}_{i}(W)\right|^{-1}\overline{w}_{i}(W)$,
$i=1,\ldots,m$, defines $\Xi$ with the required property. ∎

######  3.7. —

There exists a Borel measurable map $\Xi:\mathbb{G}(n,m)\to\mathbb{V}(n,m)$
with the property that
$W=\operatorname{\mathrm{span}}\\{\Xi_{1}(W),\ldots,\Xi_{m}(W)\\}$, for every
$W\in\mathbb{G}(n,m)$.

###### Proof.

Since $\mathbb{G}(n,m)$ is compact, it can partitioned into finitely many
Borel measurable sets $\mathscr{V}_{1},\ldots,\mathscr{V}_{J}$, each having
diameter bounded by $1/2$. Define $\Xi$, piecewise, to coincide on
$\mathscr{V}_{j}$ with $\Xi_{j}$ associated to
$\operatorname{\mathrm{Clos}}\mathscr{V}_{j}$ in 3.6, $j=1,\ldots,J$. ∎

######  3.8. —

Assume $S\subseteq\mathbb{R}^{n}$, $x_{0}\in S$, and
$\mathbf{W}_{0}:S\to\mathbb{G}(n,m)$ is Lipschitzian. There exist an open
neighborhood $U$ of $x_{0}$ in $\mathbb{R}^{n}$ and Lipschitzian maps
$\mathbf{w}_{1},\ldots,\mathbf{w}_{m},\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:U\to\mathbb{R}^{n}$
with the following properties.

  1. (1)

For every $x\in U$, the family
$\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x),\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$
is an orthonormal basis of $\mathbb{R}^{n}$.

  2. (2)

For every $x\in S\cap U$, one has

$\mathbf{W}_{0}(x)=\operatorname{\mathrm{span}}\\{\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x)\\}$

and

$\mathbf{W}_{0}(x)^{\perp}=\operatorname{\mathrm{span}}\\{\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)\\}.$

###### Proof.

We let $\widehat{\mathbf{W}}_{0}:\widehat{U}\to\mathbb{G}(n,m)$ be a
Lipschitzian extension of $\mathbf{W}_{0}$, where $\widehat{U}$ is an open
neighborhood of $S$ in $\mathbb{R}^{n}$ (recall 3.4). Abbreviate
$W_{0}:=\mathbf{W}_{0}(x_{0})$. Define
$\mathscr{V}=\mathbb{G}(n,m)\cap\left\\{W:d(W,W_{0})<1/4\right\\}$ and
$U=\widehat{U}\cap\widehat{\mathbf{W}}_{0}^{-1}(\mathscr{V})$. Apply 3.6 to
$\operatorname{\mathrm{Clos}}\mathscr{V}$ and denote by $\Xi$ the resulting
Lipschitzian map $\mathscr{V}\to(\mathbb{R}^{n})^{m}$. Next, define
$\mathscr{V}^{\perp}=\mathbb{G}(n,n-m)\cap\left\\{W^{\perp}:W\in\mathscr{V}\right\\}$,
apply 3.6 to $\operatorname{\mathrm{Clos}}\mathscr{V}^{\perp}$, and denote by
$\Xi^{\perp}$ the resulting Lipschitzian map
$\mathscr{V}^{\perp}\to(\mathbb{R}^{n})^{n-m}$. Letting
$\mathbf{w}_{i}(x)=\left(\Xi_{i}\circ\widehat{\mathbf{W}}_{0}\right)(x)$,
$i=1,\ldots,m$, and
$\mathbf{v}_{i}(x)=\left(\Xi^{\perp}_{i}\circ\widehat{\mathbf{W}}_{0}\right)(x)$,
$i=1,\ldots,n-m$, completes the proof. ∎

######  3.9. —

Assume $\mathbf{W}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$ is Borel measurable.
There exist Borel measurable maps
$\mathbf{w}_{1},\ldots,\mathbf{w}_{m},\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:\mathbb{R}^{n}\to\mathbb{R}^{n}$
with the following properties.

  1. (1)

For every $x\in\mathbb{R}^{n}$, the family
$\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x),\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$
is an orthonormal basis of $\mathbb{R}^{n}$.

  2. (2)

For every $x\in\mathbb{R}^{n}$, one has

$\mathbf{W}_{0}(x)=\operatorname{\mathrm{span}}\\{\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x)\\}$

and

$\mathbf{W}_{0}(x)^{\perp}=\operatorname{\mathrm{span}}\\{\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)\\}.$

###### Proof.

Choose $\Xi:\mathbb{G}(n,m)\to\mathbb{V}(n,m)$ and
$\Xi^{\perp}:\mathbb{G}(n,n-m)\to\mathbb{V}(n,n-m)$ as in 3.7. Letting
$\left(\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x)\right)=\left(\Xi\circ\mathbf{W}_{0}\right)(x)$
and
$\left(\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)\right)=\left(\Xi^{\perp}\circ\mathbf{W}_{0}^{\perp}\right)(x)$,
for $x\in\mathbb{R}^{n}$, completes the proof. ∎

######  3.10Definition of $\mathbf{W}(x)$. —

The typical situation that arises in the remaining part of this paper is that
we are given a set $S\subseteq\mathbb{R}^{n}$, a Lipschitzian map
$\mathbf{W}_{0}:S\to\mathbb{G}(n,m)$, and $x_{0}\in S$. We will represent
$\mathbf{W}_{0}(x)$ and $\mathbf{W}_{0}^{\perp}(x)$ in a neighborhood $U$ of
$x_{0}$ as in 3.8. We will then further reduce the size of $U$ several times,
in order that various conditions be met. With no exception, we will denote as
$\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$ the affine subspace containing $x$, of
direction $\mathbf{W}_{0}(x)$, whenever $\mathbf{W}_{0}(x)$ is defined.

######  3.11Definition of $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$ and
lower bound of its coarea factor. —

Given an open set $U\subseteq\mathbb{R}^{n}$, a Lipschitzian map
$\mathbf{v}:U\to\mathbb{R}^{n}$, and $u\in\mathbb{R}^{n}$, we define
$g_{\mathbf{v},u}:U\to\mathbb{R}$ by the formula

$g_{\mathbf{v},u}(x)=\langle\mathbf{v}(x),x-u\rangle.$

Clearly, $g_{\mathbf{v},u}$ is locally Lipschitzian. If $\mathbf{v}$ is
differentiable at $x\in U$, then so is $g_{\mathbf{v},u}$ and, for every
$h\in\mathbb{R}^{n}$, one has

$Dg_{\mathbf{v},u}(x)(h)=\langle\nabla g_{\mathbf{v},u}(x),h\rangle=\langle
D\mathbf{v}(x)(h),x-u\rangle+\langle\mathbf{v}(x),h\rangle.$ (7)

Next, we assume that we are given Lipschitzian maps
$\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:U\to\mathbb{R}^{n}$. We define
$g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}:U\to\mathbb{R}^{n-m}$ by the
formula

$g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)=\left(g_{\mathbf{v}_{1},u}(x),\ldots,g_{\mathbf{v}_{n-m},u}(x)\right).$

It is locally Lipschitzian as well. The relevance of
$g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$ stems from the following
observation, assuming that $\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}$ are
associated with $\mathbf{W}_{0}$ and $\mathbf{W}$ as in 3.8 and 3.10:

$\begin{split}u\in\mathbf{W}(x)&\iff u-x\in\mathbf{W}_{0}(x)\\\
&\iff\langle\mathbf{v}_{i}(x),u-x\rangle=0\text{, for all }i=1,\ldots,n-m\\\
&\iff g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)=0\\\ &\iff x\in
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{0\\}.\end{split}$ (8)

In fact,
$|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)|=\left|P_{\mathbf{W}_{0}(x)^{\perp}}(x-u)\right|$.

Abbreviate $g=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$. If each
$\mathbf{v}_{i}$ is differentiable at $x\in U$, and $h\in\mathbb{R}^{n}$, then

$Dg(x)(h)=\sum_{i=1}^{n-m}Dg_{\mathbf{v}_{i},u}(x)(h)e_{i},$

where, here and elsewhere, $e_{1},\ldots,e_{n-m}$ denotes the canonical basis
of $\mathbb{R}^{n-m}$. Thus, if $\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$
constitute an orthonormal family in $\mathbb{R}^{n}$, then

$Dg_{\mathbf{v}_{i},u}(x)(\mathbf{v}_{j}(x))=\delta_{i,j}+\varepsilon_{i,j}(x,u),$

where

$|\varepsilon_{i,j}(x,u)|=\left|\langle
D\mathbf{v}_{i}(x)(\mathbf{v}_{j}(x)),x-u\rangle\right|\leqslant\left(\operatorname{\mathrm{Lip}}\mathbf{v}_{i}\right)|x-u|,$
(9)

according to (7), and, in turn,

$Dg(x)(\mathbf{v}_{j}(x))=\sum_{i=1}^{n-m}\left(\delta_{i,j}+\varepsilon_{i,j}(x,u)\right)e_{i}.$

This allows for a lower bound of the coarea factor of $g$ at $x$ as follows.

$\begin{split}\left\|\wedge_{n-m}Dg(x)\right\|&\geqslant\left|Dg(x)(\mathbf{v}_{1}(x))\wedge\ldots\wedge
Dg(x)(\mathbf{v}_{n-m}(x))\right|\\\
&=\left|\left(\sum_{i=1}^{n-m}\left(\delta_{i,1}+\varepsilon_{i,1}(x,u)\right)e_{i}\right)\wedge\ldots\wedge\left(\sum_{i=1}^{n-m}\left(\delta_{i,n-m}+\varepsilon_{i,n-m}(x,u)\right)e_{i}\right)\right|\\\
&=\left|\det\left(\delta_{i,j}+\varepsilon_{i,j}(x,u)\right)_{i,j=1,\ldots,n-m}\right|.\end{split}$

In view of (9), we obtain the next lemma.

######  3.12. —

Given $\Lambda>0$ and $0<\varepsilon<1$, there exists
$\boldsymbol{\delta}_{3.12}(n,\Lambda,\varepsilon)>0$ with the following
property. Assume that

  1. (1)

$U\subseteq\mathbb{R}^{n}$ is open and $u\in\mathbb{R}^{n}$;

  2. (2)

$\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:U\to\mathbb{R}^{n}$ are Lipschitzian;

  3. (3)

$\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$ is an orthonormal family, for
every $x\in U$.

If

  1. (4)

$\operatorname{\mathrm{Lip}}\mathbf{v}_{i}\leqslant\Lambda$, for each
$i=1,\ldots,n-m$;

  2. (5)

$\operatorname{\mathrm{diam}}\left(U\cup\\{u\\}\right)\leqslant\boldsymbol{\delta}_{3.12}(n,\Lambda,\varepsilon)$;

then

$Jg_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)\geqslant 1-\varepsilon,$

at $\mathscr{L}^{n}$-almost every $x\in U$.

######  3.13Definition of $\pi_{u}$ and its relation with
$g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$. —

With $u\in\mathbb{R}^{n}$, we associate

$\pi_{u}:\mathbb{V}(n,n-m)\times\mathbb{R}^{n}\to\mathbb{R}^{n-m}:(\xi_{1},\ldots,\xi_{n-m},x)\mapsto\left(\langle\xi_{1},x-u\rangle,\ldots,\langle\xi_{n-m},x-u\rangle\right).$

When $(\xi_{1},\ldots,\xi_{n-m})\in\mathbb{V}(n,n-m)$ is fixed, we also
abbreviate as $\pi_{\xi_{1},\ldots,\xi_{n-m},u}$ the map
$\mathbb{R}^{n}\to\mathbb{R}^{n-m}$ defined by
$\pi_{\xi_{1},\ldots,\xi_{n-m},u}(x)=\pi_{u}(\xi_{1},\ldots,\xi_{n-m},x)$. It
is then rather useful to observe that, in the context described in 3.8 and
3.10, the following holds:

$\pi_{\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x),u}^{-1}\left\\{g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)\right\\}=\mathbf{W}(x).$
(10)

Indeed,

$\begin{split}h\in\mathbf{W}(x)&\iff h-x\in\mathbf{W}_{0}(x)\\\
&\iff\langle\mathbf{v}_{i}(x),h-x\rangle=0\text{, for all }i=1,\ldots,n-m\\\
&\iff\langle\mathbf{v}_{i}(x),h-u\rangle=\langle\mathbf{v}_{i}(x),x-u\rangle\text{,
for all }i=1,\ldots,n-m\\\
&\iff\pi_{\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x),u}(h)=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x).\end{split}$

In the sequel, we will sometimes abbreviate
$\xi=(\xi_{1},\ldots,\xi_{n-m})\in\mathbb{V}(n,n-m)$. It also helps to notice
that, for given $\xi\in\mathbb{V}(n,n-m)$ and $y\in\mathbb{R}^{n-m}$, the set
$\pi_{\xi,u}^{-1}\\{y\\}$ is an $m$-dimensional affine subspace of
$\mathbb{R}^{n}$.

######  3.14. —

If $B\in\mathscr{B}(\mathbb{R}^{n})$ and $u\in\mathbb{R}^{n}$, then

$h_{B}:\mathbb{V}(n,n-m)\times\mathbb{R}^{n-m}\to[0,\infty]:(\xi,y)\mapsto\mathscr{H}^{m}\left(B\cap\pi_{\xi,u}^{-1}\\{y\\}\right)$

is Borel measurable.

###### Proof.

We start by showing that, when $B$ is compact, $h_{B}$ is upper
semicontinuous. Thus, if $((\xi_{k},y_{k}))_{k}$ is a sequence in
$\mathbb{V}(n,n-m)\times\mathbb{R}^{n-m}$ that converges to $(\xi,y)$, we
ought to show that

$\mathscr{H}^{m}_{\infty}(K)\geqslant\limsup_{k}\mathscr{H}^{m}_{\infty}(K_{k}),$
(11)

where $K=B\cap\pi_{\xi,u}^{-1}\\{y\\}$ and
$K_{k}=B\cap\pi_{\xi_{k},u}^{-1}\\{y_{k}\\}$. This is indeed equivalent to the
same inequality with $\mathscr{H}^{m}_{\infty}$ replaced by $\mathscr{H}^{m}$,
according to 3.2(3) and the last sentence of 3.13. Considering, if necessary,
a subsequence of $(K_{k})_{k}$ we may assume that none of the compact sets
$K_{k}$ is empty and that the limit superior in (11) is a limit. Since the set
of nonempty compact subsets of the compact set $B$, equipped with the
Hausdorff metric, is compact, the sequence $(K_{k})_{k}$ admits a subsequence
(denoted the same way) converging to a compact set $L\subseteq B$. Given $z\in
L$, there are $z_{k}\in K_{k}$ converging to $z$. Thus,
$\pi_{u}(\xi,z)=\lim_{k}\pi_{u}(\xi_{k},z_{k})=\lim_{k}y_{k}=y$. In other
words, $z\in K$. Therefore,
$\mathscr{H}^{m}_{\infty}(K)\geqslant\mathscr{H}^{m}_{\infty}(L)$, so that
(11) follows from 3.2(1).

Next, we fix $r>0$ and we abbreviate
$\mathscr{A}=\mathscr{B}(\mathbb{R}^{n})\cap\\{B:h_{B\cap\mathbf{B}(0,r)}\text{
is Borel measurable}\\}$. Thus, we have just shown that $\mathscr{A}$ contains
the collection $\mathscr{F}(\mathbb{R}^{n})$ of all closed subsets of
$\mathbb{R}^{n}$. Observe that if $(B_{j})_{j}$ is an nondecreasing sequence
in $\mathscr{A}$ and $B=\cup_{j}B_{j}$, then
$h_{B\cap\mathbf{B}(0,r)}=\lim_{j}h_{B_{j}\cap\mathbf{B}(0,r)}$ pointwise,
thus, $B\in\mathscr{A}$. In particular, $\mathbb{R}^{n}\in\mathscr{A}$.
Furthermore, if $B,B^{\prime}\in\mathscr{A}$ and $B^{\prime}\subseteq B$, then
$h_{B\setminus B^{\prime}}=h_{B}-h_{B^{\prime}}$, because all measures
involved are finite; indeed,
$h_{B\cap\mathbf{B}(0,r)}(\xi,y)\leqslant\boldsymbol{\alpha}(m)r^{m}$, for all
$(\xi,y)$. Accordingly, $B\setminus B^{\prime}\in\mathscr{A}$. This means that
$\mathscr{A}$ is a Dynkin class. Since $\mathscr{F}(\mathbb{R}^{n})$ is a
$\pi$-system, $\mathscr{A}$ contains the $\sigma$-algebra generated by
$\mathscr{F}(\mathbb{R}^{n})$, i.e., $\mathscr{B}(\mathbb{R}^{n})$ [3, Theorem
1.6.2]. Finally, if $B\in\mathscr{B}(\mathbb{R}^{n})$, then
$h_{B}=\lim_{j}h_{B\cap\mathbf{B}(0,j)}$ pointwise, whence $h_{B}$ is Borel
measurable. ∎

######  3.15. —

Assume $B\in\mathscr{B}(\mathbb{R}^{n})$ and
$\mathbf{W}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$ is Borel measurable. The
following function is Borel measurable.

$\mathbb{R}^{n}\to[0,\infty]:x\mapsto\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right).$

###### Proof.

Let $h_{\mathbf{W},B}$ denote this function. Let
$\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be
Borel measurable maps associated with $\mathbf{W}_{0}$ in 3.9. Fix
$u\in\mathbb{R}^{n}$ arbitrarily. Define

$\Upsilon:\mathbb{R}^{n}\to\mathbb{V}(n,n-m)\times\mathbb{R}^{n-m}:x\mapsto\left(\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x),g_{\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x),u}(x)\right)$

and notice that

$h_{\mathbf{W},B}=h_{B}\circ\Upsilon$

(where $h_{B}$ is the function associated with $B$ and $u$ in 3.14), according
to (10). One notes that $\Upsilon$ is Borel measurable; whence, the conclusion
ensues from 3.14. ∎

######  3.16Definition of $\phi_{E,\mathbf{W}}$. —

Let $\mathbf{W}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$ be Borel measurable and
$E\in\mathscr{B}(\mathbb{R}^{n})$ be such that $\mathscr{L}^{n}(E)<\infty$.
For each $B\in\mathscr{B}(\mathbb{R}^{n})$, we define

$\phi_{E,\mathbf{W}}(B)=\int_{E}\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)d\mathscr{L}^{n}(x).$

This is well-defined, according to 3.15. It is easy to check that
$\phi_{E,\mathbf{W}}$ is a locally finite – hence, $\sigma$-finite – Borel
measure on $\mathbb{R}^{n}$; indeed,
$\phi_{E,\mathbf{W}}(B)\leqslant\boldsymbol{\alpha}(m)(\operatorname{\mathrm{diam}}B)^{m}\mathscr{L}^{n}(E)$.

To close this section, we discuss the relevance of $\phi_{E,\mathbf{W}}$ to
the problem of existence of “nearly Nikodým sets”.

######  3.17Definition of Nearly Nikodým set. —

Let $E\in\mathscr{B}(\mathbb{R}^{n})$. We say that $B\in\mathscr{B}(E)$ is
nearly $m$-Nikodým in $E$ if

  1. (1)

$\mathscr{L}^{n}(B)>0$;

  2. (2)

For $\mathscr{L}^{n}$-almost each $x\in E$, there is $W\in\mathbb{G}(n,m)$
such that $\mathscr{H}^{m}\left(B\cap(x+W)\right)=0$.

In case $n=2$, $m=1$, and $E=[0,1]\times[0,1]$, the existence of such a $B$
(with $\mathscr{L}^{2}(B)=1$) was established by Nikodým [12], see also [4,
Chapter 8]. For arbitrary $n\geqslant 2$ and $m=n-1$, the existence of such a
$B$ was established by Falconer [7]. In fact, in both cases, these authors
established the stronger condition that, for every $x\in B$,
$\mathscr{H}^{m}(B\cap(x+W))=0$ can be replaced by $B\cap(x+W)=\\{x\\}$. Thus,
in case $1\leqslant m<n-1$, letting $B$ be a set exhibited by Falconer, if
$x\in B$ and $W\subseteq\mathbb{G}(n,n-1)$ is such that $B\cap(x+W)=\\{x\\}$,
then picking arbitrarily $V\in\mathbb{G}(n,m)$ such that $V\subseteq W$, we
see that $B\cap(x+V)=\\{x\\}$. Whence, $B$ is also nearly $m$-Nikodým in $B$.

Assuming that $\mathbf{W}_{0}:E\to\mathbb{G}(n,m)$ is Borel measurable, we say
that $B\in\mathscr{B}(E)$ is nearly $m$-Nikodým in $E$ relative to
$\mathbf{W}$ if

  1. (1)

$\mathscr{L}^{n}(B)>0$;

  2. (2)

For $\mathscr{L}^{n}$-almost each $x\in E$, one has
$\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)=0$.

######  3.18. —

Let $E\in\mathscr{B}(\mathbb{R}^{n})$ have finite $\mathscr{L}^{n}$ measure
and $\mathbf{W}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$ be Borel measurable. The
following are equivalent.

  1. (1)

$\mathscr{L}^{n}|_{\mathscr{B}(E)}$ is absolutely continuous with respect to
$\phi_{E,\mathbf{W}}|_{\mathscr{B}(E)}$.

  2. (2)

There does not exist a nearly $m$-Nikodým set in $E$ relative to $\mathbf{W}$.

###### Proof.

A set $B\in\mathscr{B}(E)$ such that $\phi_{E,\mathbf{W}}(B)=0$ and
$\mathscr{L}^{n}(B)>0$ is, by definition, a nearly $m$-Nikodým set relative to
$\mathbf{W}$. Condition (1) is equivalent to their nonexistence. ∎

######  3.19. —

Assume that $E\in\mathscr{B}(\mathbb{R}^{n})$ and that $B\in\mathscr{B}(E)$ is
nearly $m$-Nikodým. The following hold.

  1. (1)

There exists $\mathbf{W}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$ Borel
measurable such that $B$ is nearly $m$\- Nikodým in $E$ relative to
$\mathbf{W}$.

  2. (2)

There exists $C\subseteq B$ compact and
$\overline{\mathbf{W}}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$ continuous such
that $C$ is nearly $m$-Nikodým in $C$ relative to $\overline{\mathbf{W}}$.

###### Proof.

Define a Borel measurable map
$\boldsymbol{\xi}:\mathbb{G}(n,m)\to\mathbb{V}(n,n-m)$ by
$\boldsymbol{\xi}(W)=\Xi\left(W^{\perp}\right)$, where
$\Xi:\mathbb{G}(n,n-m)\to\mathbb{V}(n,n-m)$ is as in 3.7. Choose arbitrarily
$u\in\mathbb{R}^{n}$ and define a Borel measurable map

$\Upsilon:E\times\mathbb{G}(n,m)\to\mathbb{V}(n,n-m)\times\mathbb{R}^{n-m}\\\
(x,W)\mapsto\left(\boldsymbol{\xi}(W),\langle\boldsymbol{\xi}_{1}(W),x-u\rangle,\ldots,\langle\boldsymbol{\xi}_{n-m}(W),x-u\rangle\right).$

Similarly to (10), observe that

$x+W=\pi_{\boldsymbol{\xi}(W),u}^{-1}\left\\{\left(\langle\boldsymbol{\xi}_{1}(W),x-u\rangle,\ldots,\langle\boldsymbol{\xi}_{n-m}(W),x-u\rangle\right)\right\\},$

for every $(x,W)\in E\times\mathbb{G}(n,m)$. We infer from 3.14 that

$h_{B}\circ\Upsilon:E\times\mathbb{G}(n,m)\to[0,\infty]:(x,W)\mapsto\mathscr{H}^{m}\left(B\cap(x+W)\right)$

is Borel measurable. Thus, the set

$\mathscr{E}=E\times\mathbb{G}(n,m)\cap\left\\{(x,W):\mathscr{H}^{m}\left(B\cap(x+W)\right)=0\right\\}$

is Borel measurable as well. The set
$N=E\cap\\{x:\mathscr{E}_{x}=\emptyset\\}$ is coanalytic and
$\mathscr{L}^{n}(N)=0$, by assumption. By virtue of von Neumann’s selection
theorem [13, 5.5.3], there exists a universally measurable map
$\tilde{\mathbf{W}}_{0}:E\setminus N\to\mathbb{G}(n,m)$ such that
$\tilde{\mathbf{W}}_{0}(x)\in\mathscr{E}_{x}$, for every $x\in E\setminus N$,
i.e.,
$\mathscr{H}^{m}\left(B\cap\left(x+\tilde{\mathbf{W}}_{0}(x)\right)\right)=0$.
We extend $\tilde{\mathbf{W}}_{0}$ to be an arbitrary constant on
$N\cup(\mathbb{R}^{n}\setminus E)$. This makes $\tilde{\mathbf{W}}_{0}$ an
$\mathscr{L}^{n}$-measurable map defined on $\mathbb{R}^{n}$. Therefore, it is
equal $\mathscr{L}^{n}$-almost everywhere to a Borel measurable map
$\mathbf{W}_{0}:\mathbb{R}^{n}\to\mathbb{G}(n,m)$. This proves (1).

In order to prove (2), we recall 3.4, specifically, the retraction $\rho:V\to
M_{n,m}$ and the homeomorphic identification $\varphi:\mathbb{G}(n,m)\to
M_{n,m}$. Owing to the compactness of $M_{n,m}$, there are finitely many open
balls $U_{j}$, $j=1,\ldots,J$, whose closures are contained in $V$ and whose
union contains $M_{n,m}$. Since $\mathscr{L}^{n}(B)>0$, there exists
$j\in\\{1,\ldots,J\\}$ such that $\mathscr{L}^{n}\left(B\cap E_{j}\right)>0$,
where $E_{j}=\left(\varphi\circ\mathbf{W}_{0}\right)^{-1}(U_{j})$. It follows
from Lusin’s theorem [9, 2.5.3] that there exists a compact set $C\subseteq
B\cap E_{j}$ such that $\mathscr{L}^{n}(C)>0$ and the restriction
$\mathbf{W}_{0}|_{C}$ is continuous. The map $\varphi\circ\mathbf{W}_{0}|_{C}$
takes its values in the closed ball $\operatorname{\mathrm{Clos}}U_{j}$,
therefore, it admits a continuous extension
$Y_{0}:\mathbb{R}^{n}\to\operatorname{\mathrm{Clos}}U_{j}\subseteq V$. Letting
$\overline{\mathbf{W}}_{0}=\varphi^{-1}\circ\rho\circ Y_{0}$ completes the
proof. ∎

##  4\. Common setting

######  4.1Setting for the next three sections. —

In the next three sections, we shall assume the following.

  1. (1)

$E\subseteq\mathbb{R}^{n}$ is Borel measurable and
$\mathscr{L}^{n}(E)<\infty$.

  2. (2)

$U\subseteq\mathbb{R}^{n}$ is open and $E\subseteq U$.

  3. (3)

$B\subseteq\mathbb{R}^{n}$ is Borel measurable.

  4. (4)

$\mathbf{W}_{0}:U\to\mathbb{G}(n,m)$ is Lipschitzian.

  5. (5)

$\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$, for each $x\in U$.

  6. (6)

$\Lambda>0$.

  7. (7)

$\mathbf{w}_{1},\ldots,\mathbf{w}_{m}:U\to\mathbb{R}^{n}$ and
$\operatorname{\mathrm{Lip}}\mathbf{w}_{i}\leqslant\Lambda$, $i=1,\ldots,m$.

  8. (8)

$\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:U\to\mathbb{R}^{n}$ and
$\operatorname{\mathrm{Lip}}\mathbf{v}_{i}\leqslant\Lambda$, $i=1,\ldots,n-m$.

  9. (9)

$\mathbf{W}_{0}(x)=\operatorname{\mathrm{span}}\\{\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x)\\}$,
for every $x\in U$.

  10. (10)

$\mathbf{W}_{0}(x)^{\perp}=\operatorname{\mathrm{span}}\\{\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)\\}$,
for every $x\in U$.

  11. (11)

$\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x),\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$
constitute an orthonormal basis of $\mathbb{R}^{n}$, for every $x\in U$.

##  5\. Two fibrations

######  5.1A fibered space associated with
$E,B,\mathbf{w}_{1},\ldots,\mathbf{w}_{m}$. —

We define

$F:E\times\mathbb{R}^{m}\to\mathbb{R}^{n}\times\mathbb{R}^{n}:(x,t_{1},\ldots,t_{m})\mapsto\left(x,x+\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(x)\right)$

as well as

$\Sigma=F(E\times\mathbb{R}^{m})=\mathbb{R}^{n}\times\mathbb{R}^{n}\cap\left\\{(x,u):x\in
E\text{ and }u\in\mathbf{W}(x)\right\\}.$

We will oftentimes abbreviate $(x,t)=(x,t_{1},\ldots,t_{m})$. It is obvious
that $F$ is locally Lipschitzian and, therefore, $\Sigma$ is countably
$(n+m)$-rectifiable and $\mathscr{H}^{n+m}$-measurable. We also consider the
two canonical projections

$\pi_{1}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}:(x,u)\mapsto
x\quad\text{ and
}\quad\pi_{2}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}:(x,u)\mapsto
u,$

as well as the set

$\Sigma_{B}=\Sigma\cap\pi_{2}^{-1}(B)=\mathbb{R}^{n}\times\mathbb{R}^{n}\cap\left\\{(x,u):x\in
E\text{ and }u\in B\cap\mathbf{W}(x)\right\\},$

which also is, clearly, countably $(n+m)$-rectifiable and
$\mathscr{H}^{n+m}$-measurable. With the prospect of applying the coarea
formula to $\Sigma_{B}$ and $\pi_{1}$, and to $\Sigma_{B}$ and $\pi_{2}$,
respectively, we observe that, for each fixed $x\in E$,

$\Sigma_{B}\cap\pi_{1}^{-1}\\{x\\}=\\{x\\}\times\big{(}\mathbb{R}^{n}\cap\\{u:u\in
B\cap\mathbf{W}(x)\\}\big{)},$

so that

$\mathscr{H}^{m}\left(\Sigma_{B}\cap\pi_{1}^{-1}\\{x\\}\right)=\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right),$
(12)

and that, for each fixed $u\in B$,

$\begin{split}\Sigma_{B}\cap\pi_{2}^{-1}\\{u\\}&=\big{(}\mathbb{R}^{n}\cap\\{x:x\in
E\text{ and }u\in\mathbf{W}(x)\\}\big{)}\times\\{u\\}\\\
&=\left(\mathbb{R}^{n}\cap\left\\{x:x\in E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{0\\}\right\\}\right)\times\\{u\\},\end{split}$

according to (8), so that

$\mathscr{H}^{m}\left(\Sigma_{B}\cap\pi_{2}^{-1}\\{u\\}\right)=\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{0\\}\right),$ (13)

whenever $u\in B$. It now follows from the coarea formula that

$\int_{\Sigma_{B}}J_{\Sigma}\pi_{1}d\mathscr{H}^{n+m}=\int_{E}\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)d\mathscr{L}^{n}(x)=\phi_{E,\mathbf{W}}(B)$
(14)

and

$\int_{\Sigma_{B}}J_{\Sigma}\pi_{2}d\mathscr{H}^{n+m}=\int_{B}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{0\\}\right)d\mathscr{L}^{n}(u).$
(15)

For these formulæ to be useful, we need to establish bounds for the coarea
Jacobian factors $J_{\Sigma}\pi_{1}$ and $J_{\Sigma}\pi_{2}$. In order to do
so, we notice that if $\Sigma\ni(x,u)=F(x,t)$, $F$ is differentiable at
$(x,t)$, i.e., each $\mathbf{w}_{i}$ is differentiable at $x$, $i=1,\ldots,m$,
then the approximate tangent space $T_{(x,u)}\Sigma$ exists and is generated
by the following $n+m$ vectors of $\mathbb{R}^{n}\times\mathbb{R}^{n}$:

$\begin{split}\sum_{p=1}^{n}\langle w_{j},e_{p}\rangle\frac{\partial
F}{\partial
x_{p}}(x,t)&=\left(w_{j},w_{j}+\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})\right)\,,\,j=1,\ldots,n,\\\
\frac{\partial F}{\partial
t_{k}}(x,t)&=\left(0,\mathbf{w}_{k}(x)\right)\,,\,k=1,\ldots,m,\end{split}$

where $w_{1},\ldots,w_{n}$ is an arbitrary basis of $\mathbb{R}^{n}$. As
usual, $e_{1},\ldots,e_{n}$ denotes the canonical basis of $\mathbb{R}^{n}$.

######  5.2Coarea Jacobian factor of $\pi_{1}$. —

For $\mathscr{H}^{n+m}$-almost every $(x,u)\in\Sigma$, one has

$\left(1+m^{2}\Lambda^{2}|x-u|^{2}\right)^{-\frac{m}{2}}\left(2+2m\Lambda|x-u|+m^{2}\Lambda^{2}|x-u|^{2}\right)^{-\frac{n-m}{2}}\leqslant
J_{\Sigma}\pi_{1}(x,u)\leqslant 1.$

###### Proof.

We recall 3.3. That the right hand inequality be valid follows from
$\operatorname{\mathrm{Lip}}\pi_{1}=1$. Regarding the left hand inequality,
fix $(x,u)=F(x,t)$ such that $F$ is differentiable at $(x,t)$ and let
$L:T_{(x,u)}\Sigma\to\mathbb{R}^{n}$ denote the restriction of $\pi_{1}$ to
$T_{(x,u)}\Sigma$. Define $w_{j}=\mathbf{w}_{j}(x)$, $j=1,\ldots,m$, and
$w_{j}=\mathbf{v}_{j-m}(x)$, $j=m+1,\ldots,n$. Put

$v_{j}=\sum_{p=1}^{n}\langle w_{j},e_{p}\rangle\frac{\partial F}{\partial
x_{p}}(x,t)-\frac{\partial F}{\partial
t_{j}}(x,t)=\left(w_{j},\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})\right),$

$j=1,\ldots,m$, and

$v_{j}=\sum_{p=1}^{n}\langle w_{j},e_{p}\rangle\frac{\partial F}{\partial
x_{p}}(x,t)=\left(w_{j},w_{j}+\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})\right),$

$j=m+1,\ldots,n$. Recall (6) that

$J_{\Sigma}\pi_{1}(x,u)=\|\wedge_{n}L\|\geqslant\frac{1}{|v_{1}\wedge\ldots\wedge
v_{n}|},$

since $L(v_{j})=w_{j}$, $j=1\ldots,n$. Now, notice that, for $j=1,\ldots,m$,

$\left|v_{j}\right|^{2}=|w_{j}|^{2}+\left|\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})\right|^{2}\leqslant
1+m^{2}\Lambda^{2}|t|^{2},$

whereas, for $j=m+1,\ldots,n$,

$\left|v_{j}\right|^{2}=|w_{j}|^{2}+\left|w_{j}+\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})\right|^{2}\leqslant
2+2\left|\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})\right|+\left|\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})\right|^{2}\\\
\leqslant 2+2m\Lambda|t|+m^{2}\Lambda^{2}|t|^{2}.$

Since $u=x+\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(x)$, one also has

$|u-x|^{2}=\left|\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(x)\right|^{2}=|t|^{2}.$

Finally,

$|v_{1}\wedge\ldots\wedge
v_{n}|\leqslant\left(1+m^{2}\Lambda^{2}|x-u|^{2}\right)^{\frac{m}{2}}\left(2+2m\Lambda|x-u|+m^{2}\Lambda^{2}|x-u|^{2}\right)^{\frac{n-m}{2}},$

and the conclusion follows. ∎

######  5.3. —

Let $1\leqslant q\leqslant n-1$ be an integer and let $v_{1},\ldots,v_{q}$ be
an orthonormal family in $\mathbb{R}^{n}$. There exists
$\lambda\in\Lambda(n,q)$ such that

$\left|\det\left(\langle
v_{k},e_{\lambda(j)}\rangle\right)_{j,k=1,\ldots,q}\right|\geqslant\begin{pmatrix}n\\\
q\end{pmatrix}^{-\frac{1}{2}}\,.$

Here, $\Lambda(n,q)$ denotes the set of increasing maps
$\\{1,\ldots,q\\}\to\\{1,\ldots,n\\}$.

###### Proof.

We define a linear map
$L:\mathbb{R}^{q}\to\mathbb{R}^{n}:(s_{1},\ldots,s_{q})\mapsto\sum_{k=1}^{q}s_{k}v_{k}$
and we observe that $L$ is an isometry. Therefore, its area Jacobian factor
$JL=1$, by definition. Now, also

$(JL)^{2}=\sum_{\lambda\in\Lambda(n,q)}\left|\det\left(\langle
v_{k},e_{\lambda(j)}\rangle\right)_{j,k=1,\ldots,q}\right|^{2},$

according to the Binet-Cauchy formula [6, Chapter 3 §2 Theorem 4]. The
conclusion easily follows. ∎

######  5.4Coarea Jacobian factor of $\pi_{2}$. —

The following hold.

  1. (1)

For $\mathscr{H}^{n+m}$-almost every $(x,u)\in\Sigma$, one has

$\frac{\begin{pmatrix}n\\\
n-m\end{pmatrix}^{-\frac{1}{2}}-m(n-m)\Lambda|x-u|\left(1+m\Lambda|x-u|\right)^{n-m-1}}{\left(2+2m\Lambda|x-u|+m^{2}\Lambda^{2}|x-u|^{2}\right)^{\frac{n-m}{2}}}\leqslant
J_{\Sigma}\pi_{2}(x,u)\leqslant 1.$

  2. (2)

For $\mathscr{H}^{n+m}$-almost every $(x,u)\in\Sigma$, one has
$J_{\Sigma}\pi_{2}(x,u)>0$.

###### Proof.

Clearly,
$J_{\Sigma}\pi_{2}(x,u)\leqslant(\operatorname{\mathrm{Lip}}\pi_{2})^{n}\leqslant
1$. Regarding the left hand inequality, fix $(x,u)=F(x,t)$ such that $F$ is
differentiable at $(x,t)$ and, this time, let
$L:T_{(x,u)}\Sigma\to\mathbb{R}^{n}$ denote the restriction of $\pi_{2}$ to
$T_{(x,u)}\Sigma$. We will now define a family of $n$ vectors
$v_{1},\ldots,v_{n}$ belonging to $T_{(x,u)}\Sigma$. We choose
$v_{k}=\frac{\partial F}{\partial t_{k}}(x,t)=(0,\mathbf{w}_{k}(x))$, for
$k=1,\ldots,m$. For choosing the $n-m$ remaining vectors, we proceed as
follows. We select $\lambda\in\Lambda(n,n-m)$ as in 5.3 applied with $q=n-m$
to $\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$ and we let
$v_{m+j}=\frac{\partial F}{\partial x_{\lambda(j)}}(x,t)$, $j=1,\ldots,n-m$.
Recalling (6), we have

$J_{\Sigma}\pi_{1}(x,u)=\|\wedge_{n}L\|\geqslant\frac{|L(v_{1})\wedge\ldots\wedge
L(v_{n})|}{|v_{1}\wedge\ldots\wedge v_{n}|}.$

As in the proof of 5.2, we find that

$|v_{1}\wedge\ldots\wedge
v_{n}|\leqslant\left(2+2m\Lambda|x-u|+m^{2}\Lambda^{2}|x-u|^{2}\right)^{\frac{n-m}{2}}$

and it remains only to find a lower bound for $|L(v_{1})\wedge\ldots\wedge
L(v_{n})|$. The latter equals the absolute value of the determinant of the
matrix of coefficients of $L(v_{i})$, $i=1,\ldots,n$, with respect to any
orthonormal basis of $\mathbb{R}^{n}$. We choose the basis
$\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x),\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$.
Thus,

$\begin{split}|L(v_{1})\wedge\ldots\wedge L(v_{n})|&=\left|\det\left(\begin{array}[]{c c c | c}1&\cdots&0&*\\\ \vdots&\ddots&\vdots&\vdots\\\ 0&\cdots&1&*\\\ \hline\cr 0&\cdots&0&\left\langle e_{\lambda(j)}+\sum_{i=1}^{m}t_{i}\frac{\partial\mathbf{w}_{i}}{\partial x_{\lambda(j)}}(x),\mathbf{v}_{k}(x)\right\rangle\\\ \end{array}\right)\right|\\\ &=\left|\det\left(\left\langle e_{\lambda(j)}+\sum_{i=1}^{m}t_{i}\frac{\partial\mathbf{w}_{i}}{\partial x_{\lambda(j)}}(x),\mathbf{v}_{k}(x)\right\rangle\right)_{j,k=1,\ldots,n-m}\right|.\end{split}$ (16)

Abbreviate

$h_{\lambda(j)}=\sum_{i=1}^{m}t_{i}\frac{\partial\mathbf{w}_{i}}{\partial
x_{\lambda(j)}}(x)$

and observe that $\left|h_{\lambda(j)}\right|\leqslant
m\Lambda|t|=m\Lambda|x-u|$, $j=1,\ldots,n-m$ (recall the proof of 5.2). It
remains only to remember that $\lambda$ has been selected in order that

$\left|\det\left(\left\langle
e_{\lambda(j)},\mathbf{v}_{k}(x)\right\rangle\right)_{j,k=1,\ldots,n-m}\right|\geqslant\begin{pmatrix}n\\\
n-m\end{pmatrix}^{-\frac{1}{2}}$

and to infer from the multilinearity of the determinant that

$\left|\det\left(\left\langle
e_{\lambda(j)},\mathbf{v}_{k}(x)\right\rangle+\left\langle
h_{\lambda(j)},\mathbf{v}_{k}(x)\right\rangle\right)_{j,k}-\det\left(\left\langle
e_{\lambda(j)},\mathbf{v}_{k}(x)\right\rangle\right)_{j,k}\right|\\\
\leqslant(n-m)\left(\max_{j=1,\ldots,n-m}\left|h_{\lambda(j)}\right|\right)\left(\max_{j=1,\ldots,n-m}\left|h_{\lambda(j)}\right|+\left|e_{\lambda(j)}\right|\right)^{n-m-1}\\\
\leqslant(n-m)m\Lambda|x-u|\left(1+m\Lambda|x-u|\right)^{n-m-1}.$

This completes the proof of conclusion (1).

Let $E_{0}$ denote the subset of $E$ consisting of those $x$ such that each
$\mathbf{w}_{i}$, $i=1,\ldots,m$, is differentiable at $x$. Thus, $E_{0}$ is
Borel measurable and so is

$A=E_{0}\times\mathbb{R}^{m}\cap\bigg{\\{}(x,t):\\\
\operatorname{\mathrm{rank}}\left(\begin{array}[]{c|c|c|c|c|c}\mathbf{w}_{1}(x)&\cdots&\mathbf{w}_{m}(x)&e_{1}+\sum_{i=1}^{m}t_{i}\frac{\partial\mathbf{w}_{i}}{\partial
x_{1}}(x)&\ldots&e_{n}+\sum_{i=1}^{m}t_{i}\frac{\partial\mathbf{w}_{i}}{\partial
x_{n}}(x)\end{array}\right)<n\bigg{\\}}.$

If $(x,u)\in\Sigma\setminus F(A)$, then the restriction of $\pi_{2}$ to
$T_{(x,u)}\Sigma$ is surjective and, therefore, $J_{\Sigma}\pi_{2}(x,u)>0$.
Thus, we ought to show that $\mathscr{H}^{n+m}(F(A))=0$. Since $F$ is
Lipschitzian, it suffices to establish that $\mathscr{L}^{n+m}(A)=0$. As $A$
is Borel measurable, it is enough to prove that $\mathscr{L}^{m}(A_{x})=0$,
for every $x\in E_{0}$, according to Fubini’s theorem. Fix $x\in E_{0}$. As in
the proof of conclusion (1), choose $\lambda\in\Lambda(n,n-m)$ associated with
$\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$, according to 5.3. Based on
(16), we see that

$A_{x}\subseteq\mathbb{R}^{m}\cap\left\\{t:\det\left(\left\langle
e_{\lambda(j)}+\sum_{i=1}^{m}t_{i}\frac{\partial\mathbf{w}_{i}}{\partial
x_{\lambda(j)}}(x),\mathbf{v}_{k}(x)\right\rangle\right)_{j,k=1,\ldots,n-m}=0\right\\}.$

The set on the right is of the form
$S_{x}=\mathbb{R}^{m}\cap\\{(t_{1},\ldots,t_{m}):P_{x}(t_{1},\ldots,t_{m})=0\\}$,
for some polynomial $P_{x}\in\mathbb{R}[T_{1},\ldots,T_{m}]$, and
$P_{x}(0,\ldots,0)=\det\left(\langle
e_{\lambda(j)},\mathbf{v}_{k}(x)\rangle\right)_{j,k=1,\ldots,n-m}\neq 0$. It
follows that $\mathscr{L}^{m}(S_{x})=0$ – see, e.g., [9, 2.6.5] – and the
proof of (2) is complete. ∎

######  5.5 Proposition. —

The measure $\phi_{E,\mathbf{W}}$ is absolutely continuous with respect to
$\mathscr{L}^{n}$.

###### Proof.

Let $B\in\mathscr{B}(\mathbb{R}^{n})$ be such that $\mathscr{L}^{n}(B)=0$. It
follows from (15) that

$\int_{\Sigma_{B}}J_{\Sigma}\pi_{2}d\mathscr{H}^{n+m}=0.$

It next follows from 5.4(2) that $\mathscr{H}^{n+m}(\Sigma_{B})=0$. In turn,
(14) implies that

$\phi_{E,\mathbf{W}}(B)=\int_{\Sigma_{B}}J_{\Sigma}\pi_{1}d\mathscr{H}^{n+m}=0.$

∎

######  5.6Definition of $\mathscr{Z}_{E}\mathbf{W}$. —

Note that $\phi_{E,\mathbf{W}}$ is a $\sigma$-finite Borel measure on
$\mathbb{R}^{n}$ (see 3.16) and it is absolutely continuous with respect to
$\mathscr{L}^{n}$ (see 5.5). It then ensues from the Radon-Nikodým theorem
that there exists a Borel measurable function

$\mathscr{Z}_{E}\mathbf{W}:\mathbb{R}^{n}\to\mathbb{R}$

such that, for every $B\in\mathscr{B}(\mathbb{R}^{n})$, one has

$\int_{E}\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)d\mathscr{L}^{n}(x)=\phi_{E,\mathbf{W}}(B)=\int_{B}\mathscr{Z}_{E}\mathbf{W}(u)d\mathscr{L}^{n}(u).$

Furthermore, $\mathscr{Z}_{E}\mathbf{W}$ is univoquely defined (only) up to an
$\mathscr{L}^{n}$ null set. This will not affect the reasoning in this paper.
Each time we write $\mathscr{Z}_{E}\mathbf{W}$, we mean one particular Borel
measurable function satisfying the above equality, for every
$B\in\mathscr{B}(\mathbb{R}^{n})$.

######  5.7Definition of $\mathscr{Y}_{E}^{0}\mathbf{W}$. —

We define $\mathscr{Y}^{0}_{E}\mathbf{W}:\mathbb{R}^{n}\to[0,\infty]$ by the
formula

$\mathscr{Y}_{E}^{0}\mathbf{W}(u)=\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{0\\}\right),$ (17)

$u\in\mathbb{R}^{n}$. Letting $B=\mathbb{R}^{n}$ in (13), one infers from 3.3
that $\mathscr{Y}^{0}_{E}\mathbf{W}$ is $\mathscr{L}^{n}$-measurable. Using
the estimates we have established so far regarding coarea Jacobian factors, we
now show that $\mathscr{Z}_{E}\mathbf{W}$ and $\mathscr{Y}_{E}^{0}\mathbf{W}$
are comparable, when the diameter of $E$ is not too large.

######  5.8 Proposition. —

Given $0<\varepsilon<1$, there exists
$\boldsymbol{\delta}_{5.8}(n,\Lambda,\varepsilon)>0$ with the following
property. If
$\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}_{5.8}(n,\Lambda,\varepsilon)$,
then

$(1-\varepsilon)2^{-\frac{n-m}{2}}\mathscr{Y}^{0}_{E}\mathbf{W}(u)\leqslant\mathscr{Z}_{E}\mathbf{W}(u)\leqslant(1+\varepsilon)2^{\frac{n-m}{2}}\begin{pmatrix}n\\\
n-m\end{pmatrix}^{\frac{1}{2}}\mathscr{Y}^{0}_{E}\mathbf{W}(u),$

for $\mathscr{L}^{n}$-almost every $u\in E$.

###### Proof.

We readily infer from 5.2 and 5.4(1) that there exists
$\boldsymbol{\delta}(n,\Lambda,\varepsilon)>0$ such that, for
$\mathscr{H}^{n+m}$-almost all $(x,u)\in\Sigma$, if
$|x-u|\leqslant\boldsymbol{\delta}(n,\Lambda,\varepsilon)$, then

$\alpha:=(1-\varepsilon)2^{-\frac{n-m}{2}}\leqslant J_{\Sigma}\pi_{1}(x,u)$
(18)

and

$\beta:=(1+\varepsilon)^{-1}2^{-\frac{n-m}{2}}\begin{pmatrix}n\\\
n-m\end{pmatrix}^{-\frac{1}{2}}\leqslant J_{\Sigma}\pi_{2}(x,u),$ (19)

where the above define $\alpha$ and $\beta$.

Assume now that
$\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}(n,\Lambda,\varepsilon)$.
Given $B\in\mathscr{B}(E)$, we infer from (14), 5.2, 5.4(1), (15), and the
above lower bounds, that

$\phi_{E,\mathbf{W}}(B)=\int_{\Sigma_{B}}J_{\Sigma}\pi_{1}d\mathscr{H}^{n+m}\geqslant\alpha\mathscr{H}^{n+m}(\Sigma_{B})\geqslant\alpha\int_{\Sigma_{B}}J_{\Sigma}\pi_{2}d\mathscr{H}^{n+m}\\\
=\alpha\int_{B}\mathscr{Y}^{0}_{E}\mathbf{W}d\mathscr{L}^{n}$

and

$\phi_{E,\mathbf{W}}(B)=\int_{\Sigma_{B}}J_{\Sigma}\pi_{1}d\mathscr{H}^{n+m}\leqslant\mathscr{H}^{n+m}(\Sigma_{B})\leqslant\beta^{-1}\int_{\Sigma_{B}}J_{\Sigma}\pi_{2}d\mathscr{H}^{n+m}\\\
=\beta^{-1}\int_{B}\mathscr{Y}^{0}_{E}\mathbf{W}d\mathscr{L}^{n}.$

Thus,

$\int_{B}\alpha\mathscr{Y}^{0}_{E}\mathbf{W}d\mathscr{L}^{n}\leqslant\int_{B}\mathscr{Z}_{E}\mathbf{W}d\mathscr{L}^{n}\leqslant\int_{B}\beta^{-1}\mathscr{Y}^{0}_{E}\mathbf{W}d\mathscr{L}^{n},$

for every $B\in\mathscr{B}(E)$. The conclusion follows from the
$\mathscr{L}^{n}$-measurability of both $\mathscr{Z}_{E}\mathbf{W}$ and
$\mathscr{Y}^{0}_{E}\mathbf{W}$. ∎

######  5.9Rest stop. —

The above upper bound for $\mathscr{Z}_{E}\mathbf{W}$ is already enough to
bound it from above, in turn, by a constant times
$(\operatorname{\mathrm{diam}}E)^{m}$ – see 6.4. We next want to establish
that $\mathscr{Z}_{E}\mathbf{W}>0$, almost everywhere in $E$. Yet, in the
definition (17) of $\mathscr{Y}^{0}_{E}\mathbf{W}(u)$, $u$ does not appear as
the covariable of a function whose level sets we are measuring, thereby
preventing the use of the coarea formula in an attempt to estimate
$\mathscr{Y}^{0}_{E}\mathbf{W}(u)$. This naturally leads to adding a variable
$y\in\mathbb{R}^{n-m}$ to the fibered space $\Sigma$, a covariable for
$g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$.

######  5.10A fibered space associated with
$E,B,\mathbf{w}_{1},\ldots,\mathbf{w}_{m},\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}$.
—

Let $r>0$. Abbreviate $C_{r}=\mathbb{R}^{n-m}\cap\left\\{y:|y|\leqslant
r\right\\}$, the Euclidean ball centered at the origin, of radius $r$, in
$\mathbb{R}^{n-m}$. We define

$\hat{F}_{r}:E\times\mathbb{R}^{m}\times
C_{r}\to\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n-m}:(x,t,y)\mapsto\left(x,x+\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x),y\right)$

and

$\hat{\Sigma}_{r}=\hat{F}_{r}\left(E\times\mathbb{R}^{m}\times
C_{r}\right)=\mathbb{R}^{n}\times\mathbb{R}^{n}\times
C_{r}\cap\left\\{(x,u,y):x\in E\text{ and
}u\in\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right\\}.$

We note that $\hat{F}_{r}$ is locally Lipschitzian and $\hat{\Sigma}_{r}$ is
countably $2n$-rectifiable and $\mathscr{H}^{2n}$-measurable. Similarly to
5.1, we define

$\hat{\Sigma}_{r,B}=\hat{\Sigma}_{r}\cap\pi_{2}^{-1}(B)$

which, clearly, is also countably $2n$-rectifiable and
$\mathscr{H}^{2n}$-measurable. We aim to apply the coarea formula to
$\hat{\Sigma}_{r,B}$ and to the two projections

$\pi_{1}\times\pi_{3}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n-m}\to\mathbb{R}^{n}\times\mathbb{R}^{n-m}:(x,u,y)\mapsto(x,y)$

and

$\pi_{2}\times\pi_{3}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n-m}\to\mathbb{R}^{n}\times\mathbb{R}^{n-m}:(x,u,y)\mapsto(u,y).$

To this end, we notice that

$\hat{\Sigma}_{r,B}\cap\left(\pi_{1}\times\pi_{3}\right)^{-1}\\{(x,y)\\}\\\
=\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n-m}\cap\left\\{(x,u,y):u\in
B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right\\}$

and, thus,

$\mathscr{H}^{m}\left(\hat{\Sigma}_{r,B}\cap\left(\pi_{1}\times\pi_{3}\right)^{-1}\\{(x,y)\\}\right)=\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right),$

for every $(x,y)\in E\times C_{r}$. We further notice that

$\begin{split}\hat{\Sigma}_{r,B}\cap(\pi_{2}&\times\pi_{3})^{-1}\\{(u,y)\\}\\\
&=\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n-m}\cap\left\\{(x,u,y):x\in
E\text{ and
}u\in\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right\\}\\\
&=\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n-m}\cap\left\\{(x,u,y):x\in
E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right\\},\end{split}$

because

$\begin{split}u\in\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)&\iff
u-x-\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\in\mathbf{W}_{0}(x)\\\
&\iff\left\langle\mathbf{v}_{j}(x),u-x-\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right\rangle=0\text{,
for all }j=1,\ldots,n-m\\\ &\iff
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)=y\end{split}$

and, therefore,

$\mathscr{H}^{m}\left(\hat{\Sigma}_{r,B}\cap(\pi_{2}\times\pi_{3})^{-1}\\{(u,y)\\}\right)=\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right),$

whenever $u\in B$ and $y\in C_{r}$.

It now follows from the coarea formula and Fubini’s theorem that

$\int_{\hat{\Sigma}_{r,B}}J_{\hat{\Sigma}_{r}}(\pi_{1}\times\pi_{3})d\mathscr{H}^{2n}=\int_{E}d\mathscr{L}^{n}(x)\int_{C_{r}}\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)d\mathscr{L}^{n-m}(y)$
(20)

and

$\int_{\hat{\Sigma}_{r,B}}J_{\hat{\Sigma}_{r}}(\pi_{2}\times\pi_{3})d\mathscr{H}^{2n}=\int_{B}d\mathscr{L}^{n}(u)\int_{C_{r}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y).$
(21)

######  5.11Coarea Jacobian factors of $\pi_{1}\times\pi_{3}$ and
$\pi_{2}\times\pi_{3}$. —

The following inequalities hold, for $\mathscr{H}^{2n}$-almost every
$(x,u,y)\in\hat{\Sigma}_{r}$.

$2^{-(n-m)}\left(1+2n\Lambda|u-x|+2n^{2}\Lambda^{2}|u-x|^{2}\right)^{-\frac{n}{2}}\leqslant
J_{\hat{\Sigma}_{r}}(\pi_{1}\times\pi_{3})(x,u,y)$

and

$J_{\hat{\Sigma}_{r}}(\pi_{2}\times\pi_{3})(x,u,y)\leqslant 1.$

###### Proof.

The second conclusion is obvious, since
$\operatorname{\mathrm{Lip}}\pi_{2}\times\pi_{3}=1$. Regarding the first
conclusion, we reason similarly to the proof of 5.2. Fix
$(x,u,y)=\hat{F}_{r}(x,t,y)$ such that $\hat{F}_{r}$ is differentiable at
$(x,t,y)$ and denote by $L$ the restriction of $\pi_{1}\times\pi_{3}$ to
$T_{(x,u,y)}\hat{\Sigma}_{r}$. This tangent space is generated by the
following $2n$ vectors of
$\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n-m}$:

$\begin{split}\frac{\partial\hat{F}_{r}}{\partial
x_{j}}(x,t,y)&=\left(e_{j},e_{j}+\sum_{i=1}^{m}t_{i}\frac{\partial\mathbf{w}_{i}(x)}{\partial
x_{j}}(x)+\sum_{i=1}^{n-m}y_{i}\frac{\partial\mathbf{v}_{i}}{\partial
x_{j}}(x),0\right),\,j=1,\ldots,n,\\\ \frac{\partial\hat{F}_{r}}{\partial
t_{k}}(x,t,y)&=\left(0,\mathbf{w}_{k}(x),0\right),\,k=1,\ldots,m,\\\
\frac{\partial\hat{F}_{r}}{\partial
y_{\ell}}(x,t,y)&=\left(0,\mathbf{v}_{\ell}(x),e_{\ell}\right),\,\ell=1,\ldots,n-m.\end{split}$

The range of $\pi_{1}\times\pi_{3}$ being $(2n-m)$-dimensional, we need to
select $2n-m$ vectors $v_{1},\ldots,v_{2n-m}$ in $T_{(x,u,y)}\hat{\Sigma}_{r}$
in view of obtaining a lower bound

$J_{\hat{\Sigma}_{r}}(\pi_{1}\times\pi_{3})(x,u,y)=\|\wedge_{2n-m}L\|\geqslant\frac{|L(v_{1})\wedge\ldots\wedge
L(v_{2n-m})|}{|v_{1}\wedge\ldots\wedge v_{2n-m}|}.$ (22)

Our choice of $v_{1},\ldots,v_{2n-m}$ is as follows. As in the proof of 5.2,
we let $w_{j}=\mathbf{w}_{j}(x)$, for $j=1,\ldots,m$, and
$w_{j}=\mathbf{v}_{j-m}(x)$, for $j=m+1,\ldots,n$. For $j=1,\ldots,m$, we
define

$v_{j}=\left(\sum_{p=1}^{n}\langle
w_{j},e_{p}\rangle\frac{\partial\hat{F}_{r}}{\partial
x_{p}}(x,t,y)\right)-\frac{\partial\hat{F}_{r}}{\partial t_{j}}(x,t,y)\\\
=\left(w_{j},\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})+\sum_{i=1}^{n-m}y_{i}D\mathbf{v}_{i}(x)(w_{j}),0\right),$

for $j=m+1,\ldots,n$, we define

$v_{j}=\sum_{p=1}^{m}\langle
w_{j},e_{p}\rangle\frac{\partial\hat{F}_{r}}{\partial
x_{p}}(x,t,y)=\left(w_{j},w_{j}+\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(w_{j})+\sum_{i=1}^{n-m}y_{i}D\mathbf{v}_{i}(x)(w_{j}),0\right),$

and, for $j=n+1,\ldots,2n-m$, we define

$v_{j}=\frac{\partial\hat{F}_{r}}{\partial
y_{j-n}}(x,t,y)=(0,\mathbf{v}_{j-n}(x),e_{j-n}),$

so that $L(v_{1}),\ldots,L(v_{2n-m})$ is an orthonormal basis of
$\mathbb{R}^{n}\times\mathbb{R}^{n-m}$ and, therefore, the numerator in (22)
equals 1. In order to determine an upper bound for its denominator, we start
by fixing $j=1,\ldots,n$, we abbreviate
$a_{j}(x,t,y)=\sum_{i=1}^{m}t_{i}D\mathbf{w}_{i}(x)(x)(w_{j})$ and
$b_{j}(x,t,y)=\sum_{i=1}^{n-m}y_{i}D\mathbf{v}_{i}(x)(w_{j})$, and we notice
that $|a_{j}(x,t,y)|\leqslant m\Lambda|t|\leqslant n\Lambda|t|$,
$|b_{j}(x,t,y)|\leqslant(n-m)\Lambda|y|\leqslant n\Lambda|y|$. Furthermore,
since
$u-x=\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)$,
one has $|u-x|^{2}=|t|^{2}+|y|^{2}\geqslant\max\\{|t|^{2},|y|^{2}\\}$.
Therefore, if $j=1,\ldots,m$, then

$\begin{split}|v_{j}|^{2}&=|w_{j}|^{2}+\left|a_{j}(x,t,y)+b_{j}(x,t,y)\right|^{2}\\\
&\leqslant
1+\left|a_{j}(x,t,y)\right|^{2}+\left|b_{j}(x,t,y)\right|^{2}+2\left|a_{j}(x,t,y)\right|\left|b_{j}(x,t,y)\right|\\\
&\leqslant 1+n^{2}\Lambda^{2}\left(|t|^{2}+|y|^{2}+2|t||y|\right)\\\
&\leqslant 1+2n^{2}\Lambda^{2}|u-x|^{2},\end{split}$

whereas, if $j=m+1,\ldots,n$, then

$\begin{split}\left|v_{j}\right|^{2}&=|w_{j}|^{2}+|w_{j}+a_{j}(x,t,y)+b_{j}(x,t,y)|^{2}\\\
&\leqslant 1+1+|a_{j}(x,t,y)|^{2}+|b_{j}(x,t,y)|^{2}\\\
&\quad\quad\quad+2|a_{j}(x,t,y)|+2|b_{j}(x,t,y)|+2|a_{j}(x,t,y)||b_{j}(x,t,y)|\\\
&\leqslant
2\left(1+n^{2}\Lambda^{2}|u-x|^{2}+2n\Lambda|u-x|\right),\end{split}$

and, if $j=n+1,\ldots,2n-m$, then

$\left|v_{j}\right|=\sqrt{2}.$

We conclude that

$|v_{1}\wedge\ldots\wedge v_{2n-m}|\leqslant
2^{n-m}\left(1+2n^{2}\Lambda^{2}|u-x|^{2}\right)^{\frac{m}{2}}\left(1+n^{2}\Lambda^{2}|u-x|^{2}+2n\Lambda|u-x|\right)^{\frac{n-m}{2}}\\\
\leqslant
2^{n-m}\left(1+2n^{2}\Lambda^{2}|u-x|^{2}+2n\Lambda|u-x|\right)^{\frac{n}{2}}$

and the proof is complete. ∎

######  5.12Definition of $\mathscr{Y}_{E}\mathbf{W}$. —

It follows from the coarea theorem that the function

$\mathbb{R}^{n}\times\mathbb{R}^{n-m}\to[0,\infty]:(u,y)\to\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)$

is $\mathscr{L}^{n}\otimes\mathscr{L}^{n-m}$-measurable (recall 5.10 applied
with $B=\mathbb{R}^{n}$). It now follows from Fubini’s theorem that, for each
$r>0$, the function

$\mathbb{R}^{n}\to[0,\infty]:u\mapsto\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{r}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y)$

is $\mathscr{L}^{n}$-measurable. In turn, the function

$\mathscr{Y}_{E}\mathbf{W}:\mathbb{R}^{n}\to[0,\infty]:u\mapsto\liminf_{j}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y)$

is $\mathscr{L}^{n}$-measurable. It is a replacement for
$\mathscr{Y}^{0}_{E}\mathbf{W}$ defined in 5.7. We shall establish, for
$\mathscr{Z}_{E}\mathbf{W}$, a similar lower bound to that in 5.8, this time
involving $\mathscr{Y}_{E}\mathbf{W}$. Before doing so, we notice the rather
trivial fact that if $F\subseteq E$, then

$\mathscr{Y}_{F}\mathbf{W}(u)\leqslant\mathscr{Y}_{E}\mathbf{W}(u),$

for all $u\in\mathbb{R}^{n}$.

######  5.13Preparatory remark for the proof of 5.15. —

It follows from the coarea theorem that the function

$\mathbb{R}^{n}\times\mathbb{R}^{n-m}\to[0,\infty]:(x,y)\mapsto\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)$

is $\mathscr{L}^{n}\otimes\mathscr{L}^{n-m}$-measurable (recall 5.10 applied
with $B=\mathbb{R}^{n}$). It therefore follows from Fubini’s theorem as in
5.12 that

$f_{j}:\mathbb{R}^{n}\to[0,\infty]:x\mapsto\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)d\mathscr{L}^{n-m}(y)$

is $\mathscr{L}^{n}$-measurable. Furthermore, if $B$ is bounded, then
$|f_{j}(x)|\leqslant\boldsymbol{\alpha}(m)(\operatorname{\mathrm{diam}}B)^{m}$,
for every $x\in\mathbb{R}^{n}$.

######  5.14. —

If $B$ is compact, then, for every $x\in E,$ the function

$\mathbb{R}^{n-m}\to\mathbb{R}_{+}:y\mapsto\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)$

is upper semicontinuous.

###### Proof.

The proof is analogous to that of 3.14. For each $y\in\mathbb{R}^{n-m}$,
define the compact set
$K_{y}=B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)$.
If $(y_{k})_{k}$ is a sequence converging to $y$, we ought to show that

$\mathscr{H}^{m}_{\infty}\left(K_{y}\right)\geqslant\limsup_{k}\mathscr{H}^{m}_{\infty}\left(K_{y_{k}}\right).$

Since each $K_{y}$ is a subset of an $m$-dimensional affine subspace of
$\mathbb{R}^{n}$, this is indeed equivalent to the same inequality with
$\mathscr{H}^{m}_{\infty}$ replaced by $\mathscr{H}^{m}$, according to 3.2(3).
Considering, if necessary, a subsequence of $(y_{k})_{k}$, we may assume that
none of the compact sets $K_{y_{k}}$ is empty and that the above limit
superior is a limit. Considering yet a further subsequence, we may now assume
that $(K_{y_{k}})_{k}$ converges in Hausdorff distance to some compact set
$L\subseteq B$. One checks that $L\subseteq K_{y}$. It then follows from
3.2(1) that
$\mathscr{H}^{m}_{\infty}\left(K_{y}\right)\geqslant\mathscr{H}^{m}_{\infty}(L)\geqslant\limsup_{k}\mathscr{H}^{m}_{\infty}\left(K_{y_{k}}\right)$.
∎

######  5.15 Proposition. —

Given $0<\varepsilon<1$, there exists
$\boldsymbol{\delta}_{5.15}(n,\Lambda,\varepsilon)>0$ with the following
property. If $\operatorname{\mathrm{diam}}(E\cup
B)\leqslant\boldsymbol{\delta}_{5.15}(n,\Lambda,\varepsilon)$ and $B$ is
compact, then

$\int_{E}\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)d\mathscr{L}^{n}(x)\geqslant(1-\varepsilon)2^{-(n-m)}\int_{B}\mathscr{Y}_{E}\mathbf{W}(u)d\mathscr{L}^{n}(u).$

###### Proof.

We first observe that we can choose
$\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon)>0$ small enough so
that

$J_{\hat{\Sigma}_{r}}(\pi_{1}\times\pi_{3})(x,u,y)\geqslant(1-\varepsilon)2^{-(n-m)},$
(23)

for $\mathscr{H}^{2n}$-almost every $(x,u,y)\in\hat{\Sigma}_{r}$, provided
$|u-x|\leqslant\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon)$,
according to 5.11. Thus, (23) holds, for $\mathscr{H}^{2n}$-almost every
$(x,u,y)\in\hat{\Sigma}_{r,B}$, under the assumption that
$\operatorname{\mathrm{diam}}(E\cup
B)\leqslant\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon)$. In that
case, (20), (21), and 5.11 imply that

$\begin{split}\int_{E}d\mathscr{L}^{n}(x)&\int_{C_{r}}\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)d\mathscr{L}^{n-m}(y)\\\
&=\int_{\hat{\Sigma}_{r,B}}J_{\hat{\Sigma}_{r}}(\pi_{1}\times\pi_{3})d\mathscr{H}^{2n}\\\
&\geqslant(1-\varepsilon)2^{-(n-m)}\mathscr{H}^{2n}\left(\hat{\Sigma}_{r,B}\right)\\\
&\geqslant(1-\varepsilon)2^{-(n-m)}\int_{\hat{\Sigma}_{r,B}}J_{\hat{\Sigma}_{r}}(\pi_{2}\times\pi_{3})d\mathscr{H}^{2n}\\\
&=(1-\varepsilon)2^{-(n-m)}\int_{B}d\mathscr{L}^{n}(u)\int_{C_{r}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y).\end{split}$
(24)

Fix $x\in E$ and $\beta>0$. According to 5.14, there exists a positive integer
$j(x,\beta)$ such that if $j\geqslant j(x,\beta)$, then

$\begin{split}\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)+\beta&\geqslant\sup_{y\in
C_{j^{-1}}}\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)\\\
&\geqslant\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)d\mathscr{L}^{n-m}(y).\end{split}$

Taking the limit superior as $j\to\infty$, on the right hand side, and letting
$\beta\to 0$, we obtain

$\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)\geqslant\limsup_{j}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)d\mathscr{L}^{n-m}(y).$
(25)

As this holds for all $x\in E$, we may integrate over $E$ with respect to
$\mathscr{L}^{n}$. We notice that for every $j=1,2,\ldots$,
$|f_{j}|\leqslant\boldsymbol{\alpha}(m)(\operatorname{\mathrm{diam}}B)^{m}\mathbbm{1}_{\mathbb{R}^{n}}$
(recall the notation of 5.13); the latter being $\mathscr{L}^{n}\hskip
2.5pt{\vrule height=7.0pt,width=0.5pt,depth=0.0pt}\hskip-0.2pt\vbox{\hrule
height=0.5pt,width=7.0pt,depth=0.0pt}\,E$-summable, this justifies the
application of the reverse Fatou lemma below. Thus, the following ensues from
(25), the reverse Fatou lemma, (24), and the Fatou lemma:

$\begin{split}\int_{E}\mathscr{H}^{m}&\left(B\cap\mathbf{W}(x)\right)d\mathscr{L}^{n}(x)\\\
&\geqslant\int_{E}d\mathscr{L}^{n}(x)\limsup_{j}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)d\mathscr{L}^{n-m}(y)\\\
&\geqslant\limsup_{j}\int_{E}d\mathscr{L}^{n}(x)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)\right)d\mathscr{L}^{n-m}(y)\\\
&\geqslant(1-\varepsilon)2^{-(n-m)}\limsup_{j}\int_{B}d\mathscr{L}^{n}(u)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y)\\\
&\geqslant(1-\varepsilon)2^{-(n-m)}\liminf_{j}\int_{B}d\mathscr{L}^{n}(u)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y)\\\
&\geqslant(1-\varepsilon)2^{-(n-m)}\int_{B}d\mathscr{L}^{n}(u)\liminf_{j}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C_{j^{-1}}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y)\\\
&=(1-\varepsilon)2^{-(n-m)}\int_{B}\mathscr{Y}_{E}\mathbf{W}(u)d\mathscr{L}^{n}(u).\end{split}$

∎

######  5.16 Corollary. —

If $0<\varepsilon<1$ and
$\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon)$,
then

$\mathscr{Z}_{E}\mathbf{W}(u)\geqslant(1-\varepsilon)2^{-(n-m)}\mathscr{Y}_{E}\mathbf{W}(u),$

for $\mathscr{L}^{n}$-almost every $u\in E$.

##  6\. Upper bound for $\mathscr{Y}_{E}\mathbf{W}$ and
$\mathscr{Z}_{E}\mathbf{W}$

######  6.1Bow tie lemma. —

Let $S\subseteq\mathbb{R}^{n}$, $W\in\mathbb{G}(n,m)$, and $0<\tau<1$. Assume
that

$(\forall\,x\in
S)(\forall\,0<\rho\leqslant\operatorname{\mathrm{diam}}S):S\cap\mathbf{B}(x,\rho)\subseteq\mathbf{B}(x+W,\tau\rho).$

Then there exists $F:P_{W}(S)\to\mathbb{R}^{n}$ such that
$S=\operatorname{\mathrm{im}}F$ and
$\operatorname{\mathrm{Lip}}F\leqslant\frac{1}{\sqrt{1-\tau^{2}}}$. In
particular,

$\mathscr{H}^{m}(S)\leqslant\left(\frac{1}{\sqrt{1-\tau^{2}}}\right)^{m}\boldsymbol{\alpha}(m)(\operatorname{\mathrm{diam}}S)^{m}.$

###### Proof.

Let $x,x^{\prime}\in S$ and define
$\rho=|x-x^{\prime}|\leqslant\operatorname{\mathrm{diam}}S$. Thus,
$x^{\prime}\in S\cap\mathbf{B}(x,\rho)$ and, therefore,
$\left|P_{W^{\perp}}(x-x^{\prime})\right|\leqslant\tau\rho=\tau|x-x^{\prime}|$.
Since
$|x-x^{\prime}|^{2}=\left|P_{W}(x-x^{\prime})\right|^{2}+\left|P_{W^{\perp}}(x-x^{\prime})\right|^{2}$,
we infer that

$(1-\tau^{2})\left|x-x^{\prime}\right|^{2}\leqslant\left|P_{W}(x-x^{\prime})\right|^{2}.$

Therefore, $P_{W}|_{S}$ is injective; and the Lipschitzian bound on
$F=\left(P_{W}|_{S}\right)^{-1}$ clearly follows from the above inequality.
Regarding the second conclusion, we note that

$\mathscr{H}^{m}(S)=\mathscr{H}^{m}\left(F(P_{W}(S))\right)\leqslant\left(\operatorname{\mathrm{Lip}}F\right)^{m}\mathscr{H}^{m}\left(P_{W}(S)\right)$

and $P_{W}(S)$ is contained in a ball of radius
$\operatorname{\mathrm{diam}}P_{W}(S)\leqslant\operatorname{\mathrm{diam}}S$.
∎

######  6.2. —

Given $0<\tau<1$, there exists $\boldsymbol{\delta}_{6.2}(n,\Lambda,\tau)>0$
with the following property. If

  1. (1)

$x_{0}\in U$ and $u\in\mathbb{R}^{n}$;

  2. (2)

$\operatorname{\mathrm{diam}}\left(E\cup\\{x_{0}\\}\cup\\{u\\}\right)\leqslant\boldsymbol{\delta}_{6.2}(n,\Lambda,\tau)$;

then: For each $y\in\mathbb{R}^{n-m}$, each $x\in E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}$, and each
$0<\rho<\infty$, one has

$E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\cap\mathbf{B}(x,\rho)\subseteq\mathbf{B}\left(x+\mathbf{W}_{0}(x_{0}),\tau\rho\right).$

###### Proof.

We show that
$\boldsymbol{\delta}_{\ref{ub.1}}(n,\Lambda,\tau)=\frac{\tau}{2\Lambda\sqrt{n}}$
will do. Let $x,x^{\prime}\in E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}$, for some
$y\in\mathbb{R}^{n-m}$. Thus,
$g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x^{\prime})$
and, hence,

$0=\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)-g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x^{\prime})\right|=\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x),x-u\rangle-\langle\mathbf{v}_{i}(x^{\prime}),x^{\prime}-u\rangle\right|^{2}}\\\
=\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x),x-x^{\prime}\rangle-\langle\mathbf{v}_{i}(x^{\prime})-\mathbf{v}_{i}(x),x^{\prime}-u\rangle\right|^{2}}\\\
\geqslant\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x),x-x^{\prime}\rangle\right|^{2}}-\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x^{\prime})-\mathbf{v}_{i}(x),x^{\prime}-u\rangle\right|^{2}},$

thus,

$\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x),x-x^{\prime}\rangle\right|^{2}}\leqslant\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x^{\prime})-\mathbf{v}_{i}(x),x^{\prime}-u\rangle\right|^{2}}\\\
\leqslant\sqrt{n-m}\Lambda|x-x^{\prime}||x^{\prime}-u|\leqslant\frac{\tau}{2}|x-x^{\prime}|.$

In turn,

$\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(x-x^{\prime})\right|=\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x_{0}),x-x^{\prime}\rangle\right|^{2}}\\\
\leqslant\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x^{\prime}),x-x^{\prime}\rangle\right|^{2}}+\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x^{\prime})-\mathbf{v}_{i}(x_{0}),x-x^{\prime}\rangle\right|^{2}}\\\
\leqslant\frac{\tau}{2}|x-x^{\prime}|+\sqrt{n-m}\Lambda|x^{\prime}-x_{0}||x-x^{\prime}|\leqslant\tau|x-x^{\prime}|.$

∎

######  6.3 Proposition. —

There are $\boldsymbol{\delta}_{6.3}(n,\Lambda)>0$ and
$\mathbf{c}_{6.3}(m)\geqslant 1$ with the following property. If
$u\in\mathbb{R}^{n}$ and
$\operatorname{\mathrm{diam}}(E\cup\\{u\\})\leqslant\boldsymbol{\delta}_{6.3}(n,\Lambda)$,
then

$\max\left\\{\mathscr{Y}^{0}_{E}\mathbf{W}(u),\mathscr{Y}_{E}\mathbf{W}(u)\right\\}\leqslant\mathbf{c}_{6.3}(m)(\operatorname{\mathrm{diam}}E)^{m}.$

###### Proof.

Let
$\boldsymbol{\delta}_{\ref{upper.bound}}(n,\Lambda)=\boldsymbol{\delta}_{\ref{ub.1}}(n,\Lambda,1/2)$.
Recall the definitions of $\mathscr{Y}^{0}_{E}\mathbf{W}$ and
$\mathscr{Y}_{E}\mathbf{W}$, 5.7 and 5.12, respectively. If $E=\emptyset$, the
conclusion is obvious. If not, pick $x_{0}\in E$ arbitrarily. Given any
$y\in\mathbb{R}^{n-m}$, we see that 6.2 applies with $\tau=1/2$ and, in turn,
the bow tie lemma 6.1 applies to $S=E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}$ and
$W=\mathbf{W}_{0}(x_{0})$. Thus,

$\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\\{y\\}\right)\leqslant\left(\frac{2}{\sqrt{3}}\right)^{m}\boldsymbol{\alpha}(m)(\operatorname{\mathrm{diam}}E)^{m}.$

The proposition is proved. ∎

######  6.4 Corollary. —

There are $\boldsymbol{\delta}_{6.4}(n,\Lambda)>0$ and
$\mathbf{c}_{6.4}(n)\geqslant 1$ with the following property. If
$\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}_{6.4}(n,\Lambda)$,
then

$\mathscr{Z}_{E}\mathbf{W}(u)\leqslant\mathbf{c}_{6.4}(n)(\operatorname{\mathrm{diam}}E)^{m},$

for $\mathscr{L}^{n}$-almost every $u\in E$.

###### Proof.

Let
$\boldsymbol{\delta}_{\ref{cor.ub}}(n,\Lambda)=\min\\{\boldsymbol{\delta}_{\ref{upper.bound}}(n,\Lambda),\boldsymbol{\delta}_{\ref{Z.1}}(n,\Lambda,1/2)\\}$.
∎

##  7\. Lower bound for $\mathscr{Y}_{E}\mathbf{W}$ and
$\mathscr{Z}_{E}\mathbf{W}$

######  7.1Setting for this section. —

We enforce again the exact same assumptions as in 4.1; and as in 5.10, we let
$C_{r}=\mathbb{R}^{n-m}\cap\\{y:|y|\leqslant r\\}$.

######  7.2Polyballs. —

Given $x_{0}\in\mathbb{R}^{n}$ and $r>0$, we define

$\mathbf{C}_{\mathbf{W}}(x_{0},r)=\mathbb{R}^{n}\cap\left\\{x:\left|P_{\mathbf{W}_{0}(x_{0})}(x-x_{0})\right|\leqslant
r\text{ and }\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(x-x_{0})\right|\leqslant
r\right\\}.$

We notice that, if $x\in\mathbf{C}_{\mathbf{W}}(x_{0},r)$, then
$|x-x_{0}|\leqslant r\sqrt{2}$; in particular,
$\operatorname{\mathrm{diam}}\mathbf{C}_{\mathbf{W}}(x_{0},r)\leqslant
2\sqrt{2}r$. We also notice that
$\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)=\boldsymbol{\alpha}(m)\boldsymbol{\alpha}(n-m)r^{n}$.

######  7.3. —

Given $0<\varepsilon<1/3$, there exists
$\boldsymbol{\delta}_{7.3}(n,\Lambda,\varepsilon)>0$ with the following
property. If

  1. (1)

$0<r<\boldsymbol{\delta}_{7.3}(n,\Lambda,\varepsilon)$;

  2. (2)

$u\in\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq U$;

  3. (3)

$|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0})|\leqslant(1-3\varepsilon)r$;

  4. (4)

$C\subseteq C_{\varepsilon r}$ is closed;

then

$\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\right)\geqslant\frac{1}{1+\varepsilon}\boldsymbol{\alpha}(m)r^{m}\mathscr{L}^{n-m}(C).$

######  7.4 Remark. —

With hopes that the following will help the reader form a geometrical imagery:
Under the circumstances 7.3, $\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)$ may be seen as a
“nonlinear stripe”, “horizontal” with respect to $\mathbf{W}_{0}(x_{0})$, “at
height” $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0})$ with respect to
$x_{0}$, and of “width” $C$.

###### Proof of 7.3.

Given $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$, we define

$V_{z}=\mathbb{R}^{n}\cap\left\\{x_{0}+z+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x_{0}):y\in
C_{r}\right\\}\subseteq\mathbf{C}_{\mathbf{W}}(x_{0},r)$

and we consider the isometric parametrization $\gamma_{z}:C_{r}\to V_{z}$
defined by the formula

$\gamma_{z}(y)=x_{0}+z+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x_{0}).$

We also abbreviate
$f_{z,u}=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}\circ\gamma_{z}$.

Claim #1.
$\operatorname{\mathrm{Lip}}f_{z,u}\leqslant(1+\varepsilon)^{\frac{1}{n-m}}$.

Since $\gamma_{z}$ is an isometry, it suffices to obtain an upper bound for
$\operatorname{\mathrm{Lip}}g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}|_{\mathbf{C}_{\mathbf{W}}(x_{0},r)}$.
Let $x,x^{\prime}\in\mathbf{C}_{\mathbf{W}}(x_{0},r)$ and note that

$\begin{split}\big{|}g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)&-g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x^{\prime})\big{|}=\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x),x-u\rangle-\langle\mathbf{v}_{i}(x^{\prime}),x^{\prime}-u\rangle\right|^{2}}\\\
&\leqslant\sqrt{\sum_{i=1}^{n-m}\left(\left|\langle\mathbf{v}_{i}(x)-\mathbf{v}_{i}(x^{\prime}),x-u\rangle\right|+\left|\langle\mathbf{v}_{i}(x^{\prime}),x-x^{\prime}\rangle\right|\right)^{2}}\\\
&\leqslant\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x)-\mathbf{v}_{i}(x^{\prime}),x-u\rangle\right|^{2}}+\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x^{\prime}),x-x^{\prime}\rangle\right|^{2}}\\\
&\leqslant\sqrt{n-m}\Lambda|x-x^{\prime}||x-u|+\left|P_{\mathbf{W}_{0}(x^{\prime})^{\perp}}(x-x^{\prime})\right|\\\
&\leqslant\left(1+\sqrt{n-m}\Lambda
2\sqrt{2}r\right)|x-x^{\prime}|.\end{split}$

Recalling hypothesis (1), it is now apparent that
$\boldsymbol{\delta}_{\ref{53}}$ can be chosen small enough, according to $n$,
$\Lambda$, and $\varepsilon$, so that Claim #1 holds.

Claim #2. For $\mathscr{L}^{n-m}$-almost every $y\in C_{r}$, one has
$\|Df_{z,u}(y)-\operatorname{\mathrm{id}}_{\mathbb{R}^{n-m}}\|\leqslant\varepsilon$.

Let $y\in C_{r}$ be such that $f_{z,u}$ is differentiable at $y$. We shall
estimate the coefficients of the matrix representing $Df_{z,u}(y)$ with
respect to the canonical basis. Fix $i,j=1,\ldots,n-m$ and recall (7):

$\begin{split}\frac{\partial}{\partial y_{i}}\langle
f_{u,z}(y),e_{j}\rangle&=\frac{\partial}{\partial y_{i}}\langle
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(\gamma_{z}(y)),e_{j}\rangle\\\
&=\left(\frac{\partial}{\partial
y_{i}}\right)g_{\mathbf{v}_{j},u}(\gamma_{z}(y))\\\ &=\left\langle\nabla
g_{\mathbf{v}_{j},u}(\gamma_{z}(y)),\frac{\partial\gamma_{z}(y)}{\partial
y_{i}}\right\rangle\\\ &=\left\langle
D\mathbf{v}_{j}(\gamma_{z}(y))(\mathbf{v}_{i}(x_{0})),\gamma_{z}(y)-u\right\rangle+\left\langle\mathbf{v}_{j}(\gamma_{z}(y)),\mathbf{v}_{i}(x_{0})\right\rangle\\\
&=\mathrm{I}+\mathrm{II}.\end{split}$

Next, note that

$\left|\mathrm{II}-\delta_{ij}\right|=\left|\mathrm{II}-\langle\mathbf{v}_{j}(x_{0}),\mathbf{v}_{i}(x_{0})\rangle\right|=\left|\langle\mathbf{v}_{j}(\gamma_{z}(y))-\mathbf{v}_{j}(x_{0}),\mathbf{v}_{i}(x_{0})\rangle\right|\\\
\leqslant\Lambda\left|\gamma_{z}(y)-x_{0}\right|\leqslant\Lambda
2\sqrt{2}r\leqslant\frac{\varepsilon}{2(n-m)},$

where the last inequality follows from hypothesis (1), upon choosing
$\boldsymbol{\delta}_{\ref{53}}$ small enough, according to $n$, $\Lambda$,
and $\varepsilon$. Moreover,

$\left|\mathrm{I}\right|\leqslant\Lambda|\gamma_{z}(y)-u|\leqslant\Lambda
2\sqrt{2}r\leqslant\frac{\varepsilon}{2(n-m)}.$

Therefore, if $(a_{ij})_{i,j=1,\ldots,n-m}$ is the matrix representing
$Df_{z,u}(y)$ with respect to the canonical basis, we have shown that
$|a_{ij}-\delta_{ij}|\leqslant\frac{\varepsilon}{n-m}$, for all
$i,j=1,\ldots,n-m$. This completes the proof of Claim #2.

Claim #3. $C_{\varepsilon r}\subseteq f_{z,u}(C_{r})$.

We shall show that $|y-f_{z,u}(y)|\leqslant(1-\varepsilon)r$, for every
$y\in\operatorname{\mathrm{Bdry}}C_{r}$, and the conclusion will become a
consequence of the intermediate value theorem, in case $m=n-1$, or a standard
application of homology theory, see, e.g., [5, 4.6.1], in case $m<n-1$. If
$m<n-1$, it is clearly enough to establish this inequality only for
$\mathscr{H}^{n-m-1}$-almost every $y\in\operatorname{\mathrm{Bdry}}C_{r}$. In
that case, owing to the coarea theorem [9, 3.2.22], we may choose such a $y$
so that that $f_{z,u}$ is differentiable $\mathscr{H}^{1}$-almost everywhere
on the line segment $\mathbb{R}^{n-m}\cap\\{sy:0\leqslant s\leqslant 1\\}$.
Whether $m<n-1$, or $m=n-1$, it then follows from Claim #2 that

$\left|f_{z,u}(y)-f_{z,u}(0)-y\right|=\left|\int_{0}^{1}Df_{z,u}(sy)(y)d\mathscr{L}^{1}(s)-y\right|\\\
\leqslant\int_{0}^{1}\left|Df_{z,u}(sy)(y)-y\right|d\mathscr{L}^{1}(s)\leqslant\varepsilon|y|=\varepsilon
r.$

Accordingly,

$\left|f_{z,u}(y)-y\right|\leqslant\left|f_{z,u}(y)-f_{z,u}(0)-y\right|+\left|f_{z,u}(0)\right|\leqslant\varepsilon
r+\left|f_{z,u}(0)\right|$

and the claim will be established upon showing that
$\left|f_{z,u}(0)\right|\leqslant(1-2\varepsilon)r$. Note that
$f_{z,u}(0)=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0}+z)$; we shall
use hypothesis (3) to bound its norm from above. Given $j=1,\ldots,n-m$,
recall that $\langle\mathbf{v}_{j}(x_{0}),z\rangle=0$, thus,

$\left|g_{\mathbf{v}_{j},u}(x_{0}+z)-g_{\mathbf{v}_{j},u}(x_{0})\right|=\left|\langle\mathbf{v}_{j}(x_{0}+z),x_{0}+z-u\rangle-\langle\mathbf{v}_{j}(x_{0}),x_{0}-u\rangle\right|\\\
=\left|\langle\mathbf{v}_{j}(x_{0}+z),x_{0}+z-u\rangle-\langle\mathbf{v}_{j}(x_{0}),x_{0}+z-u\rangle\right|\leqslant\Lambda|z||x_{0}+z-u|\\\
\leqslant\Lambda r2\sqrt{2}r\leqslant\frac{\varepsilon r}{\sqrt{n-m}},$

where the last inequality holds, according to hypothesis (1), provided
$\boldsymbol{\delta}_{\ref{53}}$ is chosen sufficiently small. In turn,

$\left|f_{z,u}(0)\right|\leqslant\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0}+z)-g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0})\right|+\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0})\right|\\\
\leqslant\varepsilon r+(1-3\varepsilon)r=(1-2\varepsilon)r,$

by virtue of hypothesis (3).

Claim #4. For every $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$ and every
closed $C\subseteq C_{\varepsilon r}$, one has
$\mathscr{H}^{n-m}(C)\leqslant(1+\varepsilon)\mathscr{H}^{n-m}\left(g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\cap
V_{z}\right)$.

First, notice that

$g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\cap
V_{z}=\gamma_{z}\bigg{(}\gamma_{z}^{-1}\left(g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\cap
V_{z}\right)\bigg{)}=\gamma_{z}\left(f_{z,u}^{-1}(C)\right)$

and, therefore,

$\mathscr{H}^{n-m}\left(g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\cap
V_{z}\right)=\mathscr{H}^{n-m}\left(f_{z,u}^{-1}(C)\right),$

since $\gamma_{z}$ is an isometry. Now, since $C\subseteq C_{\varepsilon
r}\subseteq f_{z,u}(C_{r})$, according to Claim #3, we have

$C=f_{z,u}\left(f_{z,u}^{-1}(C)\right).$

It therefore follows from Claim #1 that

$\begin{split}\mathscr{H}^{n-m}(C)&\leqslant\left(\operatorname{\mathrm{Lip}}f_{z,u}\right)^{n-m}\mathscr{H}^{n-m}\left(f_{z,u}^{-1}(C)\right)\\\
&\leqslant(1+\varepsilon)\mathscr{H}^{n-m}\left(g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\cap
V_{z}\right).\end{split}$

We are now ready to finish the proof, by an application of Fubini’s theorem:

$\begin{split}\mathscr{L}^{n}\bigg{(}\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap&g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\bigg{)}\\\
&=\int_{\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)}d\mathscr{L}^{m}(z)\mathscr{H}^{n-m}\left(g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\cap
V_{z}\right)\\\
&\geqslant\frac{1}{1+\varepsilon}\boldsymbol{\alpha}(m)r^{m}\mathscr{H}^{n-m}(C).\end{split}$

∎

######  7.5Lower bound for $\mathscr{Y}_{E}\mathbf{W}$. —

Given $0<\varepsilon<1/3$, there exists
$\boldsymbol{\delta}_{7.5}(n,\Lambda,\varepsilon)>0$ with the following
property. If

  1. (1)

$0<r<\boldsymbol{\delta}_{7.5}(n,\Lambda,\varepsilon)$;

  2. (2)

$\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq U$;

  3. (3)

$A\subseteq U$ is closed;

  4. (4)

$\mathscr{L}^{n}\left(A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\geqslant(1-\varepsilon)\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)$;

then

$\int_{\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)d\mathscr{L}^{n}(u)\geqslant(1-\mathbf{c}_{7.5}(n)\varepsilon)\boldsymbol{\alpha}(m)r^{m}\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right),$

where $\mathbf{c}_{7.5}(n)$ is a positive integer depending only upon $n$.

###### Proof.

Similarly to the proof of 7.3, we will first establish a lower bound for
$\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}$ on “vertical
slices” $V_{z}$ of the given polyball, followed, next, by an application of
Fubini. Given $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$, we let $V_{z}$
and $\gamma_{z}$ be as in 7.3, and we consider the set

$\check{V}_{z}=\mathbb{R}^{n}\cap\left\\{x_{0}+z+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x_{0}):y\in
C_{(1-3\varepsilon)r}\right\\}$

(notice $\check{V}_{z}$ is slightly smaller than $V_{z}$ used in the proof of
7.3) and the isometric parametrization
$\check{\gamma}_{z}:C_{(1-3\varepsilon)r}\to\check{V}_{z}$ defined by

$\check{\gamma}_{z}(y)=x_{0}+z+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x_{0}).$

For part of the proof, we find it convenient to abbreviate
$E=A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$. We also put
$\check{\mathscr{Y}}_{E}\mathbf{W}=\left(\mathscr{Y}_{E}\mathbf{W}\right)\circ\check{\gamma}_{z}$.

By definition of $\mathscr{Y}_{E}\mathbf{W}$, for each
$\check{\gamma}_{z}(y)\in\check{V}_{z}$, there exists a collection
$\mathscr{C}_{y}$ of closed balls in $\mathbb{R}^{n-m}$ with the following
properties: For every $C\in\mathscr{C}_{y}$, $C$ is a ball centered at 0,
$C\subseteq C_{\varepsilon r}$,

$\mathscr{Y}_{E}\mathbf{W}\left(\check{\gamma}_{z}(y)\right)+\varepsilon\geqslant\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{C}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},\check{\gamma}_{z}(y)}^{-1}\\{h\\}\right)d\mathscr{L}^{n-m}(h),$

and $\inf\\{\operatorname{\mathrm{diam}}C:C\in\mathscr{C}_{y}\\}=0$.
Furthermore, $\check{\mathscr{Y}}_{E}\mathbf{W}$ being
$\mathscr{L}^{n-m}$-summable, according to 6.3, there exists $N\subseteq
C_{(1-3\varepsilon)r}$ such that $\mathscr{L}^{n-m}(N)=0$ and every $y\not\in
N$ is a Lebesgue point of $\check{\mathscr{Y}}_{E}\mathbf{W}$. For such a $y$,
we may reduce $\mathscr{C}_{y}$, if necessary, keeping all the previously
stated properties valid, while enforcing also that

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{y+C}\check{\mathscr{Y}}_{E}\mathbf{W}d\mathscr{L}^{n-m}+\varepsilon\geqslant\left(\check{\mathscr{Y}}_{E}\mathbf{W}\right)(y),$

whenever $C\in\mathscr{C}_{y}$. We infer that, for each $y\in
C_{(1-3\varepsilon)r}\setminus N$ and each $C\in\mathscr{C}_{y}$,

$\int_{y+C}\check{\mathscr{Y}}_{E}\mathbf{W}d\mathscr{L}^{n-m}+2\varepsilon\mathscr{L}^{n-m}(y+C)\geqslant\int_{C}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},\check{\gamma}_{z}(y)}^{-1}\\{h\\}\right)d\mathscr{L}^{n-m}(h).$
(26)

It ensues from the Vitali covering theorem that there is a sequence
$(y_{k})_{k}$ in $C_{(1-3\varepsilon)r}\setminus N$, and
$C_{k}\in\mathscr{C}_{y_{k}}$, such that the balls $y_{k}+C_{k}$,
$k=1,2,\ldots,$ are pairwise disjoint and
$\mathscr{L}^{n-m}\left(C_{(1-3\varepsilon)r}\setminus\cup_{k=1}^{\infty}(y_{k}+C_{k})\right)=0$.
It therefore follows, from (26) and the fact that $\gamma_{z}$ is an isometry,
that

$\int_{V_{z}}\mathscr{Y}_{E}\mathbf{W}d\mathscr{H}^{n-m}+2\varepsilon\mathscr{H}^{n-m}(V_{z})\geqslant\sum_{k=1}^{\infty}\int_{C_{k}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y),$
(27)

where we have abbreviated $u_{k}=\check{\gamma}_{z}(y_{k})$. We also
abbreviate
$S_{k}=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}^{-1}(C_{k})$ and we
infer from the coarea formula that, for each $k=1,2,\ldots$,

$\int_{C_{k}}\mathscr{H}^{m}\left(E\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}^{-1}\\{y\\}\right)d\mathscr{L}^{n-m}(y)=\int_{E\cap
S_{k}}Jg_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}d\mathscr{L}^{n}\\\
\geqslant(1-\varepsilon)\mathscr{L}^{n}\left(E\cap S_{k}\right),$ (28)

where the last inequality follows from 3.12, applied with
$U=\operatorname{\mathrm{Int}}\mathbf{C}_{\mathbf{W}}(x_{0},r)$, provided that
$\boldsymbol{\delta}_{\ref{54}}(n,\Lambda,\varepsilon)$ is chosen smaller than
$(2\sqrt{2})^{-1}\boldsymbol{\delta}_{\ref{jac.g}}(n,\Lambda,\varepsilon)$.
Letting $S=\cup_{k=1}^{\infty}S_{k}$ and recalling that
$E=A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$, we infer, from (27) and (28), that

$\int_{V_{z}}\mathscr{Y}_{E}\mathbf{W}d\mathscr{H}^{n-m}+2\varepsilon\mathscr{H}^{n-m}(V_{z})\geqslant(1-\varepsilon)\mathscr{L}^{n}(E\cap
S)\\\
\geqslant(1-\varepsilon)\big{(}\mathscr{L}^{n}(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap
S)-\mathscr{L}^{n}(\mathbf{C}_{\mathbf{W}}(x_{0},r)\setminus A)\big{)}.$ (29)

Applying 7.3 to each $S_{k}$ does not immediately yield a lower bound for
$\mathscr{L}^{n}(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap S)$, because the $S_{k}$
are not necessarily pairwise disjoint. This is why we now introduce slightly
smaller versions of these:

$\check{C}_{k}=(1-\varepsilon)C_{k}\quad\text{ and
}\quad\check{S}_{k}=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}^{-1}\left(\check{C}_{k}\right).$

Claim. The sets $\check{S}_{k}\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$,
$k=1,2,\ldots$, are pairwise disjoint.

Assume, if possible, that there are $j\neq k$ and
$x\in\check{S}_{j}\cap\check{S}_{k}\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$.
Letting $\rho_{j}$ and $\rho_{k}$ denote, respectively, the radius of $C_{j}$,
and of $C_{k}$, we notice that $\rho_{j}+\rho_{k}<|y_{j}-y_{k}|$, because
$(y_{j}+C_{j})\cap(y_{k}+C_{k})=\emptyset$. Since $\check{\gamma}_{z}$ is an
isometry, we have
$|u_{j}-u_{k}|=\left|\check{\gamma}_{z}(y_{j})-\check{\gamma}_{z}(y_{k})\right|=|y_{j}-y_{k}|$
and, therefore, also

$\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{j}}(x)-g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}(x)\right|\leqslant\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{j}}(x)\right|+\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}(x)\right|\\\
\leqslant(1-\varepsilon)\rho_{j}+(1-\varepsilon)\rho_{k}<(1-\varepsilon)\left|u_{j}-u_{k}\right|.$
(30)

We now introduce the following vectors of $\mathbb{R}^{n-m}$:

$h_{j}=\sum_{i=1}^{n-m}\langle\mathbf{v}_{i}(x_{0}),u_{j}\rangle
e_{i}\quad\text{ and }\quad
h_{k}=\sum_{i=1}^{n-m}\langle\mathbf{v}_{i}(x_{0}),u_{k}\rangle e_{i}$

and we notice that

$\left|h_{j}-h_{k}\right|=\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(u_{j}-u_{k})\right|=|u_{j}-u_{k}|,$

where the second equality holds because
$u_{j}-u_{k}\in\mathbf{W}_{0}(x_{0})^{\perp}$, as clearly follows from the
definition of $\check{\gamma}_{z}$. Furthermore,

$\bigg{|}\big{(}g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{j}}(x)-g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}(x)\big{)}-\big{(}h_{k}-h_{j}\big{)}\bigg{|}=\sqrt{\sum_{i=1}^{n-m}\left|\langle\mathbf{v}_{i}(x)-\mathbf{v}_{i}(x_{0}),u_{k}-u_{j}\rangle\right|^{2}}\\\
\leqslant\sqrt{n-m}\Lambda\sqrt{2}r\left|u_{j}-u_{k}\right|\leqslant\varepsilon\left|u_{j}-u_{k}\right|,$

since we may choose $\boldsymbol{\delta}_{\ref{54}}(n,\Lambda,\varepsilon)$ to
be so small that the last inequality holds, in view of hypothesis (1). Whence,

$\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{j}}(x)-g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}(x)\right|\geqslant\left|h_{j}-h_{k}\right|-\varepsilon\left|u_{j}-u_{k}\right|=(1-\varepsilon)\left|u_{j}-u_{k}\right|,$

contradicting (30). The Claim is established.

Thus,

$\begin{split}\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap
S\right)&=\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap\cup_{k=1}^{\infty}S_{k}\right)\\\
&\geqslant\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap\cup_{k=1}^{\infty}\check{S}_{k}\right)\\\
&=\sum_{k=1}^{\infty}\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap\check{S}_{k}\right)\\\
&=\sum_{k=1}^{\infty}\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap
g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}^{-1}\left(\check{C}_{k}\right)\right)\\\
&\geqslant\frac{1}{1+\varepsilon}\boldsymbol{\alpha}(m)r^{m}\sum_{k=1}^{\infty}\mathscr{L}^{n-m}\left(\check{C}_{k}\right),\end{split}$
(31)

where the last inequality follows from 7.3. We notice that, indeed, 7.3
applies, since $\check{C}_{k}\subseteq C_{k}\subseteq C_{\varepsilon r}$ and
$\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}(x_{0})\right|=\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(u_{k}-x_{0})\right|=\left|y_{k}\right|\leqslant(1-3\varepsilon)r$.

Now,

$\sum_{k=1}^{\infty}\mathscr{L}^{n-m}\left(\check{C}_{k}\right)=(1-\varepsilon)^{n-m}\sum_{k=1}^{\infty}\mathscr{L}^{n-m}\left(C_{k}\right)=(1-\varepsilon)^{n-m}\sum_{k=1}^{\infty}\mathscr{L}^{n-m}\left(y_{k}+C_{k}\right)\\\
\geqslant(1-\varepsilon)^{n-m}\mathscr{L}^{n-m}\left(C_{(1-3\varepsilon)r}\right)\geqslant(1-3\varepsilon)^{2(n-m)}\boldsymbol{\alpha}(n-m)r^{n-m}.$
(32)

We infer, from (31) and (32), that

$\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap
S\right)\geqslant\frac{(1-3\varepsilon)^{2(n-m)}}{1+\varepsilon}\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right).$

It therefore ensues, from (29) and hypothesis (4), that

$\int_{V_{z}}\mathscr{Y}_{E}\mathbf{W}d\mathscr{H}^{n-m}+2\varepsilon\mathscr{H}^{n-m}(V_{z})\geqslant(1-\varepsilon)\left(\frac{(1-3\varepsilon)^{2(n-m)}}{1+\varepsilon}-\varepsilon\right)\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right).$

Integrating over $z$, we infer from Fubini’s theorem that

$\int_{\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}d\mathscr{L}^{n}=\int_{\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)}d\mathscr{L}^{n}(z)\int_{V_{z}}\mathscr{Y}_{E}\mathbf{W}d\mathscr{H}^{n-m}\\\
\geqslant\left[(1-\varepsilon)\left(\frac{(1-3\varepsilon)^{2(n-m)}}{1+\varepsilon}-\varepsilon\right)-2\varepsilon\right]\boldsymbol{\alpha}(m)r^{m}\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right).$

∎

######  7.6 Proposition. —

Given $0<\varepsilon<1/3$, there exist
$\boldsymbol{\delta}_{7.6}(n,\Lambda,\varepsilon)>0$ and
$\mathbf{c}_{7.6}(n)\geqslant 1$ with the following property. If

  1. (1)

$0<r<\boldsymbol{\delta}_{7.6}(n,\Lambda,\varepsilon)$;

  2. (2)

$\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq U$;

  3. (3)

$A\subseteq U$ is closed;

  4. (4)

$\mathscr{L}^{n}\left(A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\geqslant(1-\varepsilon)\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)$;

then

$\int_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)d\mathscr{L}^{n}(u)\geqslant(1-\mathbf{c}_{7.6}(n)\varepsilon)\boldsymbol{\alpha}(m)r^{m}\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right).$

######  7.7 Remark. —

The difference with 7.5 is the domain of integration (being smaller) in the
integral, on the left hand side in the conclusion.

###### Proof of 7.6.

The reader will happily check that

$\boldsymbol{\delta}_{\ref{lower.bound}}(n,\Lambda,\varepsilon)=\min\left\\{\boldsymbol{\delta}_{\ref{54}}(n,\Lambda,\varepsilon),\left(2\sqrt{2}\right)^{-1}\boldsymbol{\delta}_{\ref{upper.bound}}(n,\Lambda)\right\\}$

suits their needs. ∎

######  7.8 Proposition. —

There exists $\boldsymbol{\delta}_{7.8}(n,\Lambda)>0$ with the following
property. If
$\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}_{7.8}(n,\Lambda)$,
then

$\mathscr{Z}_{E}\mathbf{W}(u)>0,$

for $\mathscr{L}^{n}$-almost every $u\in E$.

###### Proof.

We let

$\boldsymbol{\delta}_{\ref{Z.positive}}(n,\Lambda)=\min\left\\{\boldsymbol{\delta}_{\ref{lower.bound}}\left(n,\Lambda,\frac{1}{4\mathbf{c}_{\ref{lower.bound}}(n)}\right),\boldsymbol{\delta}_{\ref{lb.2}}(n,\Lambda,1/2)\right\\}.$

According to 5.16, it suffices to show that $\mathscr{Y}_{E}\mathbf{W}(u)>0$,
for $\mathscr{L}^{n}$-almost every $u\in E$. Define
$Z=E\cap\\{u:\mathscr{Y}_{E}\mathbf{W}(u)=0\\}$ and assume, if possible, that
$\mathscr{L}^{n}(Z)>0$. Since $Z$ is $\mathscr{L}^{n}$-measurable (recall
5.12), there exists a compact set $A\subseteq Z$ such that
$\mathscr{L}^{n}(A)>0$. Observe that the sets $\mathbf{C}_{\mathbf{W}}(x,r)$,
for $x\in U$ and $r>0$, form a density basis for $\mathscr{L}^{n}$-measurable
subsets of $U$ – because their eccentricity is bounded away from zero – thus,
there exists $x_{0}\in A$ and $r_{0}>0$ such that

$\mathscr{L}^{n}\left(A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\geqslant\left(1-\frac{1}{4\mathbf{c}_{\ref{lower.bound}}(n)}\right)\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right),$

whenever $0<r<r_{0}$. There is no restriction to assume that $r_{0}$ is small
enough for $\mathbf{C}_{\mathbf{W}}(x_{0},r_{0})\subseteq U$. Thus, if we let
$r=\min\\{r_{0},\boldsymbol{\delta}_{\ref{lower.bound}}(n,\Lambda,1/(4\mathbf{c}_{\ref{lower.bound}(n)}))\\}$,
it follows from 7.6 that

$\int_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)d\mathscr{L}^{n}(u)\geqslant\left(1-\frac{1}{4}\right)\boldsymbol{\alpha}(m)r^{m}\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)>0.$
(33)

On the other hand, recalling 5.12 and the fact that
$A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq E$, we infer that
$\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)\leqslant\mathscr{Y}_{E}\mathbf{W}(u)$,
for all $u\in\mathbb{R}^{n}$. In particular,
$\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)=0$, for all
$u\in A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq Z$, contradicting (33). ∎

##  8\. Proof of the theorems

######  8.1 Theorem. —

Assume that $S\subseteq\mathbb{R}^{n}$, $\mathbf{W}_{0}:S\to\mathbb{G}(n,m)$
is Lipschitzian, and $A\subseteq S$ is Borel measurable. The following are
equivalent.

  1. (1)

$\mathscr{L}^{n}(A)=0$.

  2. (2)

For $\mathscr{L}^{n}$ almost every $x\in A$,
$\mathscr{H}^{m}(A\cap\mathbf{W}(x))=0$.

  3. (3)

For $\mathscr{L}^{n}$ almost every $x\in S$,
$\mathscr{H}^{m}(A\cap\mathbf{W}(x))=0$.

Recall our convention that $\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$.

###### Proof.

Since $\mathbb{G}(n,m)$ is complete, we can extend $\mathbf{W}_{0}$ to the
closure of $S$. Furthermore, if the theorem holds for
$\operatorname{\mathrm{Clos}}S$, then it also holds for $S$. Thus, there is no
restriction to assume that $S$ is closed.

$(1)\Rightarrow(3)$. It follows from 3.8 that each $x\in S$ admits an open
neighborhood $U_{x}$ in $\mathbb{R}^{n}$ such that $\mathbf{W}(x)$ can be
associated with a Lipschitzian orthonormal frame satisfying all the conditions
of 4.1, for some $\Lambda_{x}>0$. Since $S$ is Lindelöf, there are countably
many $x_{1},x_{2},\ldots$ such that $S\subseteq\cup_{j}U_{x_{j}}$. Letting
$E_{j}=S\cap U_{x_{j}}$, we infer from 5.5 that $\phi_{E_{j},\mathbf{W}}$ is
absolutely continuous with respect to $\mathscr{L}^{n}$. Thus, if
$\mathscr{L}^{n}(A)=0$, then
$\mathscr{H}^{m}\left(A\cap\mathbf{W}(x)\right)=0$, for
$\mathscr{L}^{n}$-almost every $x\in E_{j}$, by definition of
$\phi_{E_{j},\mathbf{W}}$. Since $j$ is arbitrary, the proof is complete.

$(3)\Rightarrow(2)$ is trivial.

$(2)\Rightarrow(1)$. Let $A$ satisfy condition (2). It is enough to show that
$\mathscr{L}^{n}(A\cap\mathbf{B}(0,r))=0$, for each $r>0$. Fix $r>0$ and
define $S_{r}=S\cap\mathbf{B}(0,r)$. Consider the $U_{x_{j}}$ defined in the
second paragraph of the present proof; since $S_{r}$ is compact, finitely many
of those, say $U_{x_{1}},\ldots,U_{x_{N}}$, cover $S_{r}$. Let
$\Lambda=\max_{j=1,\ldots,N}\Lambda_{x_{j}}$. Partition each $U_{x_{j}}$,
$j=1,\ldots,N$, into Borel measurable sets $E_{j,k}$, $k=1,\ldots,K_{j}$, such
that
$\operatorname{\mathrm{diam}}E_{j,k}\leqslant\boldsymbol{\delta}_{\ref{Z.positive}}(n,\Lambda)$.
It then follows from 7.8 that

$\left(\mathscr{Z}_{A\cap E_{j,k}}\mathbf{W}\right)(u)>0,$ (34)

for $\mathscr{L}^{n}$-almost every $u\in A\cap E_{j,k}$. Now, fix $j$ and $k$.
Observe that $\mathscr{H}^{m}\left(A\cap E_{j,k}\cap\mathbf{W}(x)\right)=0$,
for $\mathscr{L}^{n}$-almost every $x\in A\cap E_{j,k}$. Thus, $\phi_{A\cap
E_{j,k},\mathbf{W}}(A\cap E_{j,k})=0$. Moreover,

$0=\phi_{A\cap E_{j,k},\mathbf{W}}\left(A\cap E_{j,k}\right)=\int_{A\cap
E_{j,k}}\left(\mathscr{Z}_{A\cap
E_{j,k}}\mathbf{W}\right)(u)d\mathscr{L}^{n}(u).$

It follows from (34) that $\mathscr{L}^{n}(A\cap E_{j,k})=0$. Since $j$ and
$k$ are arbitrary, $\mathscr{L}^{n}(A)=0$. ∎

######  8.2 Remark. —

Alternatively, one can prove the principal implication $(2)\Rightarrow(1)$ in
two other ways. One way – more involved – consists in applying our main result
8.4 below. A second – simpler – way, along the following lines, avoids
reference to the estimates we obtained for the functions
$\mathscr{Y}_{E}\mathbf{W}$ and $\mathscr{Z}_{E}\mathbf{W}$. Consider $x\in
A$, $\eta>0$, and $V\in\mathbb{G}(n,n-m)$ such that
$d\left(V^{\perp},\mathbf{W}_{0}(x)\right)<\eta$, and put $V_{x}=x+V$. Define

$\Phi:(U_{x}\cap
V_{x})\times\mathbb{R}^{m}\to\mathbb{R}^{n}:(\xi,t)\mapsto\xi+\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(\xi)\,.$

$\Phi$ is locally Lipschitzian and one checks that, in fact, $\Phi$ is a
lipeomorphism between $B$ and $\Phi(B)$, where $B=\mathbf{B}(x,\rho)$, for
some $\rho>0$ depending upon $\eta$ and $\Lambda_{x}$, because its
differential is close to the identity. Referring to Fubini’s theorem, one then
further checks that

$\mathscr{L}^{n}(A^{\prime})=0\text{ if and only if
}\mathscr{H}^{m}(A^{\prime}\cap\mathbf{W}(\xi))=0\text{, for
$\mathscr{H}^{n-m}$-almost every }\xi\in V_{x}.$

Finally, using Fubini again, with respect to the decomposition
$\mathbb{R}^{n}=\mathbf{W}_{0}(x)\oplus\mathbf{W}_{0}(x)^{\perp}$, one shows
that $\mathscr{H}^{m}(A\cap\mathbf{W}(\zeta))=0$, for
$\mathscr{H}^{n-m}$-almost every $\zeta\in
x^{\prime}+\mathbf{W}_{0}(x)^{\perp}$, where $x^{\prime}$ is as close as we
wish to $x$. Applying the previous construction with $x^{\prime}$ replacing
$x$, using $V=\mathbf{W}_{0}(x)^{\perp}$, we find that
$\mathscr{L}^{n}(A\cap\mathbf{B}(x,r))=0$, for some $r>0$ depending on
$\Lambda_{x}$.

Notwithstanding, it seems that the (simpler) change of variable described here
is not enough to yield the (stronger) theorem below.

######  8.3Polyballs. —

Recalling 7.2, we notice that

$\mathbf{C}_{\mathbf{W}}(x_{0},r)=\mathbb{R}^{n}\cap\left\\{\nu_{x_{0}}(x-x_{0})\leqslant
r\right\\},$

where $\nu_{x_{0}}$ is a norm on $\mathbb{R}^{n}$ defined by the formula

$\nu_{x_{0}}(x)=\max\left\\{\left|P_{\mathbf{W}_{0}(x_{0})}(x)\right|,\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(x)\right|\right\\},$

for $x\in\mathbb{R}^{n}$. It is readily observed that
$\operatorname{\mathrm{Lip}}\nu_{x_{0}}\leqslant 1$.

  1. (1)

One has $|\nabla\nu_{x_{0}}(x)|=1$, for $\mathscr{L}^{n}$-almost every
$x\in\mathbb{R}^{n}$.

Abbreviate $P=P_{\mathbf{W}_{0}(x_{0})}$ and
$Q=P_{\mathbf{W}_{0}(x_{0})^{\perp}}$ and define
$S=\mathbb{R}^{n}\cap\\{x:|P(x)|=|Q(x)|\\}$, so that $\mathscr{L}^{n}(S)=0$.
Let $x\in\mathbb{R}^{n}\setminus S$ and notice $\nu_{x_{0}}$ is differentiable
at $x$. We henceforth assume that $|P(x)|<|Q(x)|$, whence,
$\nu_{x_{0}}(x)=|Q(x)|$ – the proof in the other case is similar. Define
$\varepsilon=|Q(x)|-|P(x)|>0$, $e=\frac{Q(x)}{|Q(x)|}$, and let
$0<t<\varepsilon$. Note that
$|Q(x+te)|=|Q(x)|\left(1+\frac{t}{|Q(x)|}\right)>|Q(x)|$ and
$|P(x+te)|\leqslant|P(x)|+t<|Q(x)$. Thus, $\nu_{x_{0}}(x+te)=|Q(x+te)|$ and,
in turn, $\nu_{x_{0}}(x+te)-\nu_{x_{0}}(x)=|Q(x+te)|-|Q(x)|=t$. It follows
that $\langle\nabla\nu_{x_{0}}(x),e\rangle=1$. Since $e$ is a unit vector and
$\operatorname{\mathrm{Lip}}\nu_{x_{0}}\leqslant 1$, we conclude that
$|\nabla\nu_{x_{0}}(x)|=1$.

  1. (2)

Let $U$, $\mathbf{W}$ and $\Lambda>0$ be as in 4.1. Assume
$\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq U$,
$x\in\mathbf{C}_{\mathbf{W}}(x_{0},r)$, and $0\leqslant t\leqslant 1$ is so
that $\nu_{x_{0}}(x-x_{0})=tr$. It follows that

$\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap\mathbf{W}(x)\subseteq\mathbf{B}\left(x,r(1+t)+8m\Lambda
r^{2}\right)\cap\mathbf{W}(x).$

First notice that $|z|\leqslant\sqrt{2}\nu_{x_{0}}(z)$, for every
$z\in\mathbb{R}^{n}$, so that
$|x-x_{0}|\leqslant\sqrt{2}\nu_{x_{0}}(x-x_{0})=\sqrt{2}tr$ and, for every
$x^{\prime}\in\mathbf{C}_{\mathbf{W}}(x_{0},r)$,
$|x-x^{\prime}|\leqslant\sqrt{2}\left(\nu_{x_{0}}(x-x_{0})+\nu_{x_{0}}(x_{0}-x^{\prime})\right)\leqslant\sqrt{2}r(1+t)$.
Next notice that, for each $h\in\mathbb{R}^{n}$,

$\left|\left(P_{\mathbf{W}_{0}(x_{0})}-P_{\mathbf{W}_{0}(x)}\right)(h)\right|=\left|\sum_{i=1}^{m}\langle\mathbf{w}_{i}(x_{0}),h\rangle\mathbf{w}_{i}(x_{0})-\sum_{i=1}^{m}\langle\mathbf{w}_{i}(x),h\rangle\mathbf{w}_{i}(x)\right|\\\
\leqslant\sum_{i=1}^{m}\left|\langle\mathbf{w}_{i}(x_{0}),h\rangle\right|\left|\mathbf{w}_{i}(x_{0})-\mathbf{w}_{i}(x)\right|+\sum_{i=1}^{m}\left|\langle\mathbf{w}_{i}(x_{0})-\mathbf{w}_{i}(x),h\rangle\right|\left|\mathbf{w}_{i}(x)\right|\\\
\leqslant 2m\Lambda\left|x-x_{0}\right||h|.$

We now assume that
$x^{\prime}\in\mathbf{C}_{\mathbf{W}}(x,r)\cap\mathbf{W}(x)$, in particular,
$x-x^{\prime}\in\mathbf{W}_{0}(x)$, whence,

$\left|x-x^{\prime}\right|=\left|P_{\mathbf{W}_{0}(x)}(x-x^{\prime})\right|\leqslant\left|P_{\mathbf{W}_{0}(x_{0})}(x-x^{\prime})\right|+\left|\left(P_{\mathbf{W}_{0}(x_{0})}-P_{\mathbf{W}_{0}(x)}\right)(x-x^{\prime})\right|\\\
\leqslant\nu_{x_{0}}(x-x_{0})+\nu_{x_{0}}(x^{\prime}-x_{0})+2m\Lambda|x-x_{0}||x-x^{\prime}|\leqslant
r(1+t)+4m\Lambda t(1+t)r^{2}.$

######  8.4 Theorem. —

Assume that $A\subseteq\mathbb{R}^{n}$ is Borel measurable and that
$\mathbf{W}_{0}:A\to\mathbb{G}(n,m)$ is Lipschitzian. It follows that

$\limsup_{r\to
0^{+}}\frac{\mathscr{H}^{m}\left(A\cap\mathbf{B}(x,r)\cap\mathbf{W}(x)\right)}{\boldsymbol{\alpha}(m)r^{m}}\geqslant\frac{1}{2^{n}},$

for $\mathscr{L}^{n}$-almost every $x\in A$.

Recall our convention that $\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$.

###### Proof.

Extend $\mathbf{W}_{0}$ to $\mathbb{R}^{n}$ in a Borel measurable way, for
instance, to be an arbitrary constant outside of $A$. It follows from 3.15
that, for each $r>0$, the function
$\mathbb{R}^{n}\to[0,\infty]:x\mapsto\frac{\mathscr{H}^{m}(A\cap\mathbf{B}(x,r)\cap\mathbf{W}(x))}{\boldsymbol{\alpha}(m)r^{m}}$
is Borel measurable. Thus, for each $j=1,2,\ldots$, the function
$g_{j}:\mathbb{R}^{n}\to[0,\infty]$ defined by

$g_{j}(x)=\sup_{0<r<\frac{1}{j}}\frac{\mathscr{H}^{m}\left(A\cap\mathbf{B}(x,r)\cap\mathbf{W}(x)\right)}{\boldsymbol{\alpha}(m)r^{m}}=\sup_{\begin{subarray}{c}0<r<\frac{1}{j}\\\
r\,\text{rational}\end{subarray}}\frac{\mathscr{H}^{m}\left(A\cap\mathbf{B}(x,r)\cap\mathbf{W}(x)\right)}{\boldsymbol{\alpha}(m)r^{m}}$

is Borel measurable as well, and so is $g=\lim_{j}g_{j}=\inf_{j}g_{j}$.

Abbreviate $\boldsymbol{\eta}(n,m)=2^{-(n-m)}$. Arguing reductio ad absurdum,
we henceforth assume that $A$ and $\mathbf{W}_{0}$ fail the conclusion of the
theorem. Thus, the set
$Z_{0}=A\cap\left\\{x:g(x)<\frac{\boldsymbol{\eta}(n,m)}{2^{m}}\right\\}$ is
Borel measurable and non Lebesgue null. Accordingly, there exists
$\varepsilon_{1}>0$ such that the set
$Z_{0}^{\prime}=A\cap\left\\{x:g(x)<(1-\varepsilon_{1})\frac{\boldsymbol{\eta}(n,m)}{(2+\varepsilon_{1})^{m}}\right\\}$
is also Borel measurable and of positive Lebesgue measure. It therefore ensues
from Egoroff’s theorem [9, 2.3.7] that there exists a closed set $Z\subseteq
Z^{\prime}_{0}\subseteq A$ such that $\mathscr{L}^{n}(Z)>0$ and that there
exists a positive integer $j_{0}$ such that

$\frac{\mathscr{H}^{m}\left(Z\cap\mathbf{B}(x,r)\cap\mathbf{W}(x)\right)}{\boldsymbol{\alpha}(m)r^{m}}\leqslant
g_{j_{0}}(x)<(1-\varepsilon_{1})\frac{\boldsymbol{\eta}(n,m)}{(2+\varepsilon_{1})^{m}},$
(35)

for each $x\in Z$ and each $0<r<\frac{1}{j_{0}}$. Choose
$0<\varepsilon_{2}<10$ such that

$\frac{1-\varepsilon_{1}}{1-\varepsilon_{2}}\leqslant
1-\frac{\varepsilon_{1}}{2}$ (36)

and choose $0<\varepsilon_{3}<1/3$ such that

$1-\frac{\varepsilon_{1}}{2}<1-\mathbf{c}_{\ref{lower.bound}}(n)\varepsilon_{3}.$
(37)

As in the proof of 7.8, we recall that the family
$\mathbf{C}_{\mathbf{W}}(x,r)$, for $x\in\mathbb{R}^{n}$ and $r>0$, is a
density basis of $\mathscr{L}^{n}$-measurable sets. Since
$\mathscr{L}^{n}(Z)>0$, there exists $x_{0}\in Z$ such that

$\lim_{r\to
0^{+}}\frac{\mathscr{L}^{n}\left(Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)}{\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)}=1.$

In particular, there exists $R>0$ such that

$(1-\varepsilon_{3})\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\leqslant\mathscr{L}^{n}\left(Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right),$
(38)

whenever $0<r<R$.

We let $U$ be an open neighborhood of $x_{0}$ in $\mathbb{R}^{n}$ associated
with $A$ and $\mathbf{W}_{0}$ in 3.8, so that $\mathbf{W}_{0}$ and
$\mathbf{W}_{0}^{\perp}$ are associated with orthonormal frames as in 4.1, for
some $\Lambda>0$. Define

$r_{0}=\min\bigg{\\{}1,\frac{1}{j_{0}(2+8m\Lambda)},\frac{\varepsilon_{1}}{8m\Lambda},\frac{\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon_{2})}{2\sqrt{2}},\frac{\operatorname{\mathrm{dist}}(x_{0},\mathbb{R}^{n}\setminus
U)}{2\sqrt{2}},\boldsymbol{\delta}_{\ref{lower.bound}}(n,\Lambda,\varepsilon_{3}),R\bigg{\\}}.$

Let $0<r<r_{0}$ and observe that

$\begin{split}\left(1-\mathbf{c}_{\ref{lower.bound}}(n)\varepsilon_{3}\right)&\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\leqslant\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\frac{\mathscr{Y}_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)}{\boldsymbol{\alpha}(m)r^{m}}d\mathscr{L}^{n}(u)\intertext{(by
\ref{lower.bound} applied with $\varepsilon=\varepsilon_{3}$ and
$A=Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$)}&\leqslant\frac{1}{(1-\varepsilon_{2})\boldsymbol{\eta}(n,m)}\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\frac{\mathscr{H}^{m}\left(Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap\mathbf{W}(x)\right)}{\boldsymbol{\alpha}(m)r^{m}}d\mathscr{L}^{n}(x)\end{split}$
(39)

(by 5.15, applied with $\varepsilon=\varepsilon_{2}$ and
$E=B=Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$). We also note that

$\begin{split}&\int_{Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\frac{\mathscr{H}^{m}\left(Z\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap\mathbf{W}(x)\right)}{\boldsymbol{\alpha}(m)r^{m}}d\mathscr{L}^{n}(x)\\\
&=\int_{Z\cap\\{\nu_{x_{0}}\leqslant