# On the volume of the Minkowski sum of zonoids

Matthieu Fradelizi, Mokshay Madiman, Mathieu Meyer and Artem Zvavitch
supported in part by the Agence Nationale de la Recherche, project GeMeCoD
(ANR 2011 BS01 007 01).supported in part by the U.S. National Science
Foundation through grants DMS-1409504.supported in part by the U.S. National
Science Foundation Grant DMS-1101636, Simons Foundation, the CNRS and the
Bézout Labex.

###### Abstract

We explore some inequalities in convex geometry restricted to the class of
zonoids. We show the equivalence, in the class of zonoids, between a local
Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-
submodularity of volume, and the Dembo-Cover-Thomas conjecture on the
monotonicity of the ratio of volume to the surface area. In addition to these
equivalences, we confirm these conjectures in ${\mathbb{R}}^{3}$ and we
establish an improved inequality in ${\mathbb{R}}^{2}$. Along the way, we give
a negative answer to a question of Adam Marcus regarding the roots of the
Steiner polynomial of zonoids. We also investigate analogous questions in the
$L_{p}$-Brunn-Minkowski theory, and in particular, we confirm all of the above
conjectures in the case $p=2$, in any dimension.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on mixed volumes
  3. 3 Equivalent forms of the log-submodularity of volume
    1. 3.1 Submodularity
    2. 3.2 Classes closed under sums and dilation
    3. 3.3 Classes closed under linear transformations
  4. 4 Some Special Cases
    1. 4.1 An improved inequality in ${\mathbb{R}}^{2}$
    2. 4.2 Parallelotopes in general dimension
  5. 5 Inequalities for zonoids
    1. 5.1 Zonoids in ${\mathbb{R}}^{3}$
    2. 5.2 More equivalent formulations for zonoids
  6. 6 Inequalities for $L_{p}$-zonoids
    1. 6.1 The case $p=2$
    2. 6.2 The case $p\neq 2$

##  1 Introduction

The deep parallels between inequalities in Information Theory and Convex
Geometry have been explored intensively. The recognition of these parallels
goes back to at least 1984, when Costa and Cover [14] observed the formal
resemblance between the Brunn-Minkowski inequality in Convex Geometry and the
entropy power inequality in Information Theory. The Brunn-Minkowski inequality
(see, e.g., [26, 44]) states that for all compact sets $A,B$ in
${\mathbb{R}}^{n}$, $|A+B|^{1/n}\geq|A|^{1/n}+|B|^{1/n}$, where we write $|A|$
for the volume of $A$. The entropy power inequality (see, e.g., [16]) states
that for any pair of independent random vectors $X,Y$ in ${\mathbb{R}}^{n}$,
$N(X+Y)\geq N(X)+N(Y)$, where $N(X)=e^{\frac{2h(X)}{n}}$ denotes the entropy
power of $X$, and $h(X)=-\int f(x)\log f(x)dx$ is the entropy of $X$ if $X$
has density $f$ (and $h(X)=-\infty$ if not). Despite the apparent distinctness
of the settings, a compelling case can be made (see, e.g., [17, 46, 26, 48,
36]) that these inequalities are deeply connected – in particular, it is now
well understood that the functional $A\mapsto|A|^{1/n}$ in the geometry of
compact subsets of ${\mathbb{R}}^{n}$, and the functional $f_{X}\mapsto N(X)$
in probability are analogous in many (but not all) ways. In the last decade,
several further developments have been made that link Information Theory to
the Brunn-Minkowski theory, including entropy analogues of Blaschke-Santaló’s
inequality [33], reverse Brunn-Minkowski’s inequality [6, 7], Rogers-
Shephard’s inequality [9, 35] and Busemann’s inequality [4]. Indeed, volume
inequalities and entropy inequalities (and also certain small ball
inequalities [37]) can be unified using the framework of Rényi entropies; this
framework and the relevant literature is surveyed in [36].

The analogies between Information Theory and Convex Geometry are not so direct
when one moves to second-order functionals. In particular, convex geometry
analogues of Fisher information inequalities hold sometimes but not always
[22, 2, 25]. Motivated by these analogies, Dembo, Cover and Thomas [17]
proposed the inequality, called concavity of the ratio of volume to the
surface area,

$\frac{|A+B|}{|\partial(A+B)|}\geq\frac{|A|}{|\partial
A|}+\frac{|B|}{|\partial B|}$ (1)

as a natural analogue dual to the Fisher information inequality. It was
noticed already in [17] (see also [27]) that (1) is true when $A$ (or $B$) is
an Euclidean Ball, moreover, in [22], it was proved that this last inequality
holds for any convex bodies $A$ and $B$ in ${\mathbb{R}}^{2}$. It is
interesting to note that even a weaker conjecture, called monotonicity of the
ratio of volume to the surface area,

$\frac{|A+B|}{|\partial(A+B)|}\geq\frac{|A|}{|\partial A|}$ (2)

is not trivial. The case of $B$ being a segment already gives another natural
conjecture

$\frac{|\partial(P_{u^{\perp}}A)|}{|P_{u^{\perp}}A|}\leq\frac{|\partial
A|}{|A|},$ (3)

where $P_{u^{\perp}}A$ denotes the orthogonal projection $A$ onto the
hyperplane with normal vector $u\in S^{n-1}$. We explain in details the
relationships between the above conjectures as parts of Theorems 3.6 and 3.9
below. It was proved in [27] (see also Lemma 5.3 for a simple proof) that (3)
holds with with a multiplicative constant:

$\frac{|\partial(P_{u^{\perp}}A)|}{|P_{u^{\perp}}A|}\leq\frac{2(n-1)}{n}\frac{|\partial
A|}{|A|},$ (4)

Moreover, it was shown in [22] that the constant is sharp. Thus, there are
counterexamples to (3) in ${\mathbb{R}}^{n}$ for any $n\geq 3$ and
inequalities (1) and (2) are not true in general for the whole class of convex
bodies in ${\mathbb{R}}^{n}$, $n\geq 3$. Conjecture (1) is connected with the
following conjectured inequality for the volume of projections:

$\frac{|A+B|}{|P_{u^{\bot}}(A+B)|_{n-1}}\geq\frac{|A|}{|P_{u^{\bot}}A|_{n-1}}+\frac{|B|}{|P_{u^{\bot}}B|_{n-1}}.$
(5)

Bonnesen proved in [11] (see [44] equation (7.196)) that, for any convex
bodies $A$, $B$ in ${\mathbb{R}}^{n}$,

$|A+B|\geq\left(|P_{u^{\bot}}A|_{n-1}^{\frac{1}{n-1}}+|P_{u^{\bot}}B|_{n-1}^{\frac{1}{n-1}}\right)^{n-1}\left(\frac{|A|}{|P_{u^{\bot}}A|_{n-1}}+\frac{|B|}{|P_{u^{\bot}}B|_{n-1}}\right),$
(6)

which is (5) for $n=2$. The fact that (1) is true in dimension $2$ and in the
case when one of the bodies is an Euclidean ball inspires the natural
conjecture that (1) holds for zonoids. Recall that zonoids are Hausdorff
limits of zonotopes and that zonotopes are Minkowski sums of segments. Our
main goal, in this paper, is to study a weaker version of this conjecture: is
it true that for two zonoids $A,B$ in ${\mathbb{R}}^{n}$ we have

$\frac{|A+B|}{|P_{u^{\bot}}(A+B)|_{n-1}}\geq\frac{|A|}{|P_{u^{\bot}}A|_{n-1}}?$
(7)

Note that (7) is again not true for general convex bodies and $n\geq 3$. We
prove it for zonoids and $n=3$. We also present a number of equivalent and
useful restatements of (7).

The second main contribution of this paper has to do with an analogue of the
Plünnecke-Ruzsa [40, 42] inequality for zonoids, or equivalently, the log-
submodularity property of volume on the space of zonoids with respect to
Minkowski summation. More precisely, we study the following conjecture from
[24]: given zonoids $A,B_{1},B_{2}$ in ${\mathbb{R}}^{n}$, one has

$|A|\,|A+B_{1}+B_{2}|\,\leq\,|A+B_{1}|\,|A+B_{2}|.$ (8)

This conjecture was inspired by Bobkov and the second-named author [7] who
proved that for convex bodies $A,B_{1},B_{2}$ in ${\mathbb{R}}^{n}$, one has

$|A|\,|A+B_{1}+B_{2}|\leq 3^{n}|A+B_{1}|\,|A+B_{2}|.$

Recently, it was proved in [24] that the constant $3^{n}$ in the preceding
inequality may be replaced by $\varphi^{n}$, where $\varphi=(1+\sqrt{5})/2$ is
the golden ratio, that the best constant $c_{n}$ is lower bounded by
$(4/3+o(1))^{n}$ and that $c_{2}=1$ and $c_{3}=4/3$. These observations for
dimensions $n\geq 3$ imply that, for general convex bodies, it is impossible
to have (8). In this paper, we prove that (8) holds in ${\mathbb{R}}^{3}$ for
zonoids.

This paper is organized as follows. In Section 2, we collect background
material on mixed volumes. Numerous equivalent descriptions of log-
submodularity, on a given class of convex bodies, are explored in Section 3.
In Section 4, we first present an improved version of $\log$-submodularity in
${\mathbb{R}}^{2}$ by proving a strong version of two dimensional conjecture
of T. Courtade, and we then prove inequality (7) in the special case of
paralleletopes. Section 5 explores log-submodularity of volume on the class of
zonoids– we first prove that this holds in ${\mathbb{R}}^{3}$, and then
discuss an array of inequalities that are equivalent to log-submodularity for
zonoids in arbitrary dimension (all of which therefore now hold in
${\mathbb{R}}^{3}$). Along the way, we answer a question of Adam Marcus about
Steiner polynomials of zonoids. Finally, in Section 6, inspired by a work of
Brazitikos and McIntyre [12] on vector-valued Maclaurin inequalities, we
discuss possible extensions of our results to the more general class of
$L^{p}$-zonoids, which appear in the more general $L_{p}$-Brunn-Minkowski
theory.

Acknowledgments. We are indebted to Shiri Artstein-Avidan, Guillaume Aubrun,
Silouanos Brazitikos, Dan Florentin, Dylan Langharst, Ivan Soprunov, and Ramon
Van Handel, for a number of valuable discussions and suggestions.

##  2 Preliminaries on mixed volumes

In this section, we introduce basic notations and collect essential facts and
definitions from convex geometry that are used in the paper. As a general
reference on the theory, we use [44]. We write $\langle x,y\rangle$ for the
inner product of vectors $x$ and $y$ in ${\mathbb{R}}^{n}$ and by $|x|$ the
Euclidean norm of a vector $x\in{\mathbb{R}}^{n}$. The closed Euclidean ball
in ${\mathbb{R}}^{n}$ is denoted by $B_{2}^{n}$, and its boundary by
$S^{n-1}$. We also denote by $e_{1},\dots,e_{n}$ the standard orthonormal
basis in ${\mathbb{R}}^{n}$. Moreover, for any set in
$K\subset{\mathbb{R}}^{n}$, we denote its boundary by $\partial K$. A compact
set $K$ in ${\mathbb{R}}^{n}$ is called star-shaped if, for every $x\in K$,
the segment $[0,x]$ is a subset of $K$; its radial function $\rho_{K}$ is then
defined by $\rho_{K}(x)=\sup\\{a;ax\in K\\}$. When $0$ belongs to the interior
of $K$ then $\|x\|_{K}=\rho_{K}^{-1}(x)$ is the Minkowski functional of $K$. A
convex body is a convex, compact set with non-empty interior. For a convex
body $K$, we define its support function by $h_{K}(x)=\max_{y\in K}\langle
x,y\rangle$.

We write $|K|_{m}$ for the $m$-dimensional Lebesgue measure (volume) of a
measurable set $K\subset{\mathbb{R}}^{n}$, where $m=1,...,n$ is the dimension
of the minimal affine space containing $K$, we often use the shorten notation
$|K|$ for $n$-dimensional volume. From [44, Theorem 5.1.6], for any compact
convex sets $K_{1},\dots,K_{r}$ in ${\mathbb{R}}^{n}$ and any non-negative
numbers $t_{1},\dots,t_{r}$, one has

$\displaystyle\left|t_{1}K_{1}+\cdots+t_{r}K_{r}\right|=\sum_{i_{1},\dots,i_{n}=1}^{r}t_{i_{1}}\cdots
t_{i_{n}}V(K_{i_{1}},\dots,K_{i_{n}})$ (9)

for some non-negative numbers $V(K_{i_{1}},\dots,K_{i_{n}})$, which are the
mixed volumes of $K_{i_{1}},\dots,K_{i_{n}}$. We also often use a two bodies
version of (9):

$\displaystyle\left|A+tB\right|=\sum_{k=0}^{n}{n\choose
k}t^{k}V(A[n-k],B[k]),$ (10)

for any $t>0$ and compact, convex sets $A,B$ in ${\mathbb{R}}^{n}$, where for
simplicity we use notation $A[m]$ for a convex set $A$ repeated $m$ times.
Mixed volumes satisfy a number of extremely useful inequalities. The first one
is the Brunn-Minkowski inequality $|K+L|^{1/n}\geq|K|^{1/n}+|L|^{1/n},$
whenever $K,L$ and $K+L$ are measurable. A direct consequence are Minkowski’s
first inequality

$\displaystyle V(L,K[n-1])\geq|L|^{1/n}|K|^{(n-1)/n},$ (11)

and Minkowski’s second inequality

$\displaystyle V(L,K[n-1])^{2}\geq|K|V(L[2],K[n-2]),$ (12)

for two convex, compact subsets $K$ and $L$ in ${\mathbb{R}}^{n}$. We will use
the classical integral representation for the mixed volume:

$V(L,K[n-1])=\frac{1}{n}\int_{S^{n-1}}h_{L}(u)dS_{K}(u),$ (13)

where $S_{K}$ is the surface area measure of $K$ [44]. Mixed volumes are also
useful to study the volume of the orthogonal projections of convex bodies. Let
$E$ be an $m$-dimensional subspace of ${\mathbb{R}}^{n}$, for $1\leq m\leq n$
and let $P_{E}:{\mathbb{R}}^{n}\to E$ be the orthogonal projection onto $E$.
Then for any convex body $K$ we have

$\displaystyle|U|_{n-m}|P_{E}K|_{m}={{n}\choose{m}}V(K[m],U[n-m]),$ (14)

where $U$ is any convex body in the subspace $E^{\perp}$ orthogonal to $E$. It
follows from (14) that for any orthonormal system $u_{1},\dots,u_{r},$ $1\leq
r\leq n$ we get

$\left|P_{[u_{1},\dots,u_{r}]^{\bot}}K\right|_{n-r}=\frac{n!}{(n-r)!}V(K[n-r],[0,u_{1}],\dots,[0,u_{r}]).$
(15)

For example, denote by $u^{\perp}=\\{x\in{\mathbb{R}}^{n}:\langle
x,u\rangle=0\\}$ the hyperplane orthogonal to a vector $u\in S^{n-1}$, we
obtain

$\displaystyle|P_{u^{\perp}}K|_{n-1}=nV(K[n-1],[0,u]).$ (16)

Another useful formula is connected with the computation of surface area and
mixed volumes:

$\displaystyle|\partial K|=nV(K[n-1],B_{2}^{n}),$ (17)

where by $|\partial K|$ we denote the surface area of the convex body $K$ in
${\mathbb{R}}^{n}$.

A polytope which is the Minkowski sum of finitely many line segments is called
a zonotope. Limits of zonotopes in the Hausdorff metric are called zonoids,
see [44, Section 3.2] for details. Consider zonotopes
$Z_{j}=\sum_{i_{j}=1}^{N_{j}}[0,w_{i_{j}j}]$, where, $j=1,\dots,n$ and
$w_{i_{j}j}\in{\mathbb{R}}^{n}$. Using the linearity of mixed volumes, we get
that

$V(Z_{1},\dots,Z_{n})=\sum
V([0,w_{i_{1}1}],\dots[0,w_{i_{n}n}])=\frac{1}{n!}\sum|{\mathop{\rm
det}}(\\{w_{i_{j}j}\\}_{j=1}^{n})|,$ (18)

where the sums runs over the integers $i_{j}\in\\{1,\dots,N_{j}\\}$, for
$j\in\\{1,\dots,n\\}$. Notice that, if $Z_{1}=[0,u]$, with $u\in S^{n-1}$, we
may use the basic properties of determinants to show that

$V([0,u],Z_{2},\dots,Z_{n})=\frac{1}{n!}\sum|{\mathop{\rm
det}}(\\{P_{u^{\perp}}w_{i_{j}j}\\}_{j=2}^{n})|,$

where the sum is taken over all integer $i_{j}\in\\{1,\dots,N_{j}\\}$, for
$j\in\\{2,\dots,n\\}$. We mainly use the following particular case: for
$Z=[0,u_{1}]+\cdots+[0,u_{m}]$, then

$\displaystyle|Z|=\\!\\!\\!\sum_{1\leq i_{1}<\cdots<i_{n}\leq m}$
$\displaystyle|{\mathop{\rm det}}(u_{i_{1}},\dots,u_{i_{n}})|$ (19)
$\displaystyle|P_{e^{\bot}}Z|=\\!\\!\\!\sum_{1\leq i_{2}<\cdots<i_{n}\leq
m}|{\mathop{\rm det}}(P_{e^{\bot}}u_{i_{2}},\dots,P_{e^{\bot}}u_{i_{n}})|.$

##  3 Equivalent forms of the log-submodularity of volume

###  3.1 Submodularity

Let us recall the notion of submodular set function and remind some basic
properties, see [34, 47, 21, 24] for more information on this subject and the
proofs of the theorems. We denote ${[n]}=\\{1,\cdots,n\\}$ and let $2^{[n]}$
be the family of subsets of ${[n]}$.

######  Definition 3.1.

A set function $F:2^{[n]}\rightarrow{\mathbb{R}}$ is submodular if

$\displaystyle F(S\cup T)+F(S\cap T)\leq F(S)+F(T)\quad\mbox{for all subsets
$S,T$ of $[n]$}.$ (20)

Submodularity is closely related to a partial ordering on hypergraphs [18, 10,
3]. Let $\mbox{${\cal M}$}(n,m)$ be the following family of multi-hypergraphs:
each consists of non-empty subsets $S_{i}$ of $[n]$, $\sum_{i}|S_{i}|=m$, with
$S_{i}=S_{j}$ allowed. Consider a given multi-hypergraph
$\mathcal{C}=\\{S_{1},\dots,S_{l}\\}\in\mbox{${\cal M}$}(n,m)$. Take any pair
of non-nested sets $\\{S_{i},S_{j}\\}\subset\mathcal{C}$ and let
$\mathcal{C}^{\prime}=\mathcal{C}(i,j)$ be obtained from $\mathcal{C}$ by
replacing $S_{i}$ and $S_{j}$ by $S_{i}\cap S_{j}$ and $S_{i}\cup S_{j}$,
keeping only $S_{i}\cup S_{j}$ if $S_{i}\cap S_{j}=\emptyset$.
$\mathcal{C}^{\prime}$ is called an elementary compression of $\mathcal{C}$.
The result of a sequence of elementary compressions is called a compression.
Define a partial order on $\mbox{${\cal M}$}(n,m)$ by setting
$\mathcal{A}>\mathcal{A}^{\prime}$ if $\mathcal{A}^{\prime}$ is a compression
of $\mathcal{A}$. Using transitivity of the partial order and reducing to
elementary compression, we get the following theorem (see [3]).

######  Theorem 3.2.

Suppose $F$ is a submodular function on $[n]$. Let $\mathcal{A}$ and
$\mathcal{B}$ be finite multi-hypergraph of subsets of $[n]$, with
$\mathcal{A}>\mathcal{B}$. Then

$\displaystyle\sum_{S\in\mathcal{A}}F(S)\geq\sum_{T\in\mathcal{B}}F(T).$

For every multi-hypergraph $\mathcal{A}\in\mbox{${\cal M}$}(n,m)$ there is a
unique minimal multi-hypergraph $\mathcal{A}^{\\#}$ dominated by $\mathcal{A}$
consisting of the sets $S^{\\#}_{j}=\\{i\in[n]:i\text{ lies in at least
}j\text{ of the sets }S\in\mathcal{A}\\}$. One implication of Theorem 3.2 is
that submodular functions $F$ with $F(\varphi)=0$ are “fractionally
subadditive” (see, e.g., [38])– this property has also been investigated in
connection with volumes of Minkowski sums [8, 23, 5].

We also have a notion of submodularity on the positive octant of the Euclidean
space.

######  Definition 3.3.

A function $f:{\mathbb{R}}_{+}^{n}\rightarrow{\mathbb{R}}$ is submodular if,
for any $x,y\in\mathbb{R}_{+}^{n}$,

$\displaystyle f(x\vee y)+f(x\wedge y)\leq f(x)+f(y),$

where $x\vee y$ (resp. $x\wedge y$) denotes the componentwise maximum (resp.
minimum) of $x$ and $y$.

The next very simple lemma connects submodularity for functions defined on
${\mathbb{R}}_{+}^{n}$ to submodularity for set functions. For a set
$S\subset\\{1,\dots,n\\}$, let $\hbox{\rm\large{1}}_{S}$ be the vector in
${\mathbb{R}}^{n}$ such that for each $i\in\\{1,\dots,n\\}$, the $i$-th
coordinate of $\hbox{\rm\large{1}}_{S}$ is 1, for $i\in S$ and $0$, for
$i\notin S$.

######  Lemma 3.4.

If $f:{\mathbb{R}}_{+}^{n}\rightarrow{\mathbb{R}}$ is submodular, and we set
$F(S):=f(\hbox{\rm\large{1}}_{S})$ for each $S\subset\\{1,\dots,n\\}$, then
$F$ is a submodular set function.

The fact that submodular functions are closely related to functions with
decreasing differences is classical (see, e.g., [34] or [47], which describes
more general results involving arbitrary lattices). We denote by
$\partial^{2}_{i,j}$ the second derivative with respect to coordinates $i,j$.

######  Proposition 3.5.

Let $f:{\mathbb{R}}_{+}^{n}\rightarrow{\mathbb{R}}$ be a $C^{2}$ function.
Then $f$ is submodular if and only if $\partial_{i,j}^{2}f\leq 0$ on
${\mathbb{R}}_{+}^{n}$, for every $i\neq j$.

###  3.2 Classes closed under sums and dilation

Let us now show the various forms that the notions of submodularity and log-
submodularity can have applied to volume of convex compact sets.

######  Theorem 3.6.

Consider a collection $\cal{K}$ of compact convex sets in ${\mathbb{R}}^{n}$
stable by sums and dilation. Then the following statements are equivalent:

  1. 1.

For every $m\geq 1$ and every $A,B_{1},\dots,B_{m}$ in ${\cal K}$

$\displaystyle|A|^{m-1}\left|A+\sum_{i=1}^{m}B_{i}\right|\leq\prod_{i=1}^{m}|A+B_{i}|.$
(21)

  2. 2.

For every $m\geq 1$ and $A,B_{1},\dots,B_{m}\in\cal{K}$, the function
$\bar{w}:2^{[m]}\rightarrow[0,\infty)$ defined by

$\displaystyle\bar{w}(S)=\log\bigg{|}A+\sum_{i\in S}B_{i}\bigg{|}$ (22)

for each $S\in 2^{[m]}$, is a submodular.

  3. 3.

For every $m\geq 1$ and any multi-hypergraphs $\mathcal{A},\mathcal{B}$ on
$[m]$ with $\mathcal{A}>\mathcal{B}$,

$\displaystyle\sum_{S\in\mathcal{A}}\bar{w}(S)\geq\sum_{T\in\mathcal{B}}\bar{w}(T),$

where $\bar{w}$ is defined by (22).

  4. 4.

For every $m\geq 1$ and every $A,B_{1},\dots,B_{m}\in\cal{K}$, the function
$w:{\mathbb{R}}_{+}^{m}\rightarrow[0,\infty)$ defined, for
$x=(x_{1},\dots,x_{m})\in{\mathbb{R}}_{+}^{m}$, by

$\displaystyle w(x)=\log\bigg{|}A+\sum_{i=1}^{m}x_{i}B_{i}\bigg{|},$ (23)

is submodular.

  5. 5.

For every $A,B_{1},B_{2}\in\cal{K}$

$\displaystyle|A|V(A[n-2],B_{1},B_{2})\leq\frac{n}{n-1}V(A[n-1],B_{1})V(A[n-1],B_{2}).$

###### Proof.

2\. $\implies$ 1\. This follows directly from the definition.
1\. $\implies$ 2.: We use (21) with $A+\sum_{i\in S\cap T}B_{i}$, $\sum_{i\in
S\setminus T}B_{i}$ and $\sum_{i\in T\setminus S}B_{i}$.
2\. $\Longleftrightarrow$ 3.: This follows from Theorem 3.2 applied to
$-\bar{w}$.
2\. $\Longleftrightarrow$ 4.: This comes from Lemma 3.4 and the fact that
$\cal{C}$ is dilation invariant.
4\. $\Longleftrightarrow$ 5.: We apply Proposition 3.5 to $f=-w$. Let
$K_{x}=A+\sum_{i}x_{i}B_{i}$ and $v(x)=|K_{x}|$. Then

$\displaystyle\partial_{j}v(x)=\lim_{\varepsilon\rightarrow
0}\frac{|K_{x}+\varepsilon B_{j}|-|K_{x}|}{\varepsilon}=nV(K_{x}[n-1],B_{j}),$

and for $k\neq j$

$\displaystyle\begin{split}\partial_{j,k}^{2}v(x)&=n\partial_{k}V(K_{x}[n-1],B_{j})\\\
&=n\lim_{\varepsilon\rightarrow 0}\frac{V((K_{x}+\varepsilon
B_{k})[n-1],B_{j})-V(K_{x}[n-1],B_{j})}{\varepsilon}\\\
&=n(n-1)V(K_{x}[n-2],B_{k},B_{j}).\end{split}$

By Proposition 3.5, the submodularity of $w$ is equivalent to
$\partial_{j,k}^{2}w(x)\leq 0$. But $w=\log(v)$ so

$\displaystyle\partial_{j,k}^{2}w(x)=\frac{v\partial_{jk}v-\partial_{j}v\partial_{k}v}{v^{2}},$

which is negative if and only if

$\displaystyle|K_{x}|V(K_{x}[n-2],B_{j},B_{k})\leq\frac{n}{n-1}V(K_{x}[n-1],B_{j})V(K_{x}[n-1],B_{k}).$

Thus plugging $x=0$ we get $4.\implies 5.$ and using 4. with $K$ instead of
$A$ gives $5.\implies 4.$ $\Box$

We note that it is enough to check property 1. in Theorem 3.6, just in the
case $m=2$. Indeed the case of general $m>2$ follows by an iteration argument.

We also note that, in dimension $2$, property 5. of Theorem 3.6 holds for any
convex bodies by the classical local version of Alexandrov’s inequality that
was proved by W. Fenchel (see [19], also [43] and discussion in Section 4.1
below) and further generalized in [22, 2, 45]: for any convex compact sets
$A,B_{1},B_{2}$ in ${\mathbb{R}}^{n}$ we have

$|A|V(A[n-2],B_{1},B_{2})\leq 2V(A[n-1],B_{1})V(A[n-1],B_{2}).$ (24)

The constant $2$ is sharp in any dimension (see [27] and [24]). This shows
also that log-submodularity doesn’t hold in the set of compact convex sets in
${\mathbb{R}}^{n}$, for $n\geq 3$. From (24) and Theorem 3.6, the following
theorem holds.

######  Theorem 3.7.

For every $A,B_{1},B_{2}$ convex compact in ${\mathbb{R}}^{2}$ it holds

$|A|\,|A+B_{1}+B_{2}|\leq|A+B_{1}|\,|A+B_{2}|.$

It is easy to see that 1. from Theorem 3.6 works well with direct sums, more
precisely if $A^{1},B^{1}_{1},\dots,B_{m}^{1}\subset{\mathbb{R}}^{n_{1}}$ and
$A^{2},B^{2}_{1},\dots,B_{m}^{2}\subset{\mathbb{R}}^{n_{2}}$ satisfy (21),
then $A=A^{1}\times A^{2}\subset{\mathbb{R}}^{n_{1}+n_{2}}$ and
$B_{i}=B^{1}_{i}\times B_{i}^{2}$ in ${\mathbb{R}}^{n_{1}+n_{2}}$,
$i=1,\dots,m$ satisfies (21). This fact can be used to create different
classes of compact convex sets stable by sum and dilations which satisfy the
properties of Theorem 3.6.

######  Remark 3.8.

Minkowski’s second inequality gives that, for any compact convex sets $A$ and
$B$, one has

$\displaystyle|A|V(A[n-2],B[2])\leq V(A[n-1],B)^{2}.$ (25)

Theorem 3.6 implies that if $\mathcal{K}$ is a class of convex bodies on which
log-submodularity holds (property 1. of Theorem 3.6), then for bodies in
$\mathcal{K}$, a Fenchel type inequality similar to (24) holds with a
dimensional factor $\frac{n}{n-1}$ instead of $2$.

Let us, also, note that the same proof shows that for a fixed compact convex
set $A$ not necessarily belonging to $\cal{K}$, if, for every $B_{1},B_{2}$ in
${\cal K}$, one has

$\displaystyle|A|\,|A+B_{1}+B_{2}|\leq|A+B_{1}|\,|A+B_{2}|,$ (26)

then, for every $B_{1},B_{2}\in\cal{K}$, one has

$\displaystyle|A|V(A[n-2],B_{1},B_{2})\leq\frac{n}{n-1}V(A[n-1],B_{1})V(A[n-1],B_{2}).$

###  3.3 Classes closed under linear transformations

######  Theorem 3.9.

Let ${\cal L}$ be a class of a compact convex sets in ${\mathbb{R}}^{n}$
stable under any linear transformations. The following are equivalent.

  1. 1.

$|A|\,|\partial(A+[0,u])|\leq|\partial A|\,|A+[0,u]|$ for any $A\in{\cal L}$
and any $u\in{\mathbb{R}}^{n}$.

  2. 2.

$|A|\,|\partial(P_{u^{\bot}}A)|\leq|\partial A|\,|P_{u^{\bot}}A|_{n-1}$, for
any $A\in{\cal L}$ and any $u\in S^{n-1}$.

  3. 3.

$|A|\,|P_{[u,v]^{\bot}}A|_{n-2}\sqrt{1-\langle
u,v\rangle^{2}}\leq|P_{u^{\bot}}A|_{n-1}|P_{v^{\bot}}A|_{n-1}$, for any
$A\in{\cal L}$ and any $u,v\in S^{n-1}$.

  4. 4.

$|A+[0,u]+[0,v]|\,|A|\leq|A+[0,u]|\,|A+[0,v]|$ for any $A\in{\cal L}$ and any
$u,v\in{\mathbb{R}}^{n}$.

  5. 5.

$|A|V(A[n-2],Z_{1},Z_{2})\leq\frac{n}{n-1}V(A[n-1],Z_{1})V(A[n-1],Z_{2})$, for
any $A\in{\cal L}$ and any $Z_{1},Z_{2}$ zonoids.

  6. 6.

For any $A\in{\cal L}$ and any $u,v\in{\mathbb{R}}^{n}$, let us define
$P(t)=|A+t([0,u]+[0,v])|$, for $t>0$. Then $P(t)$ is the restriction to
${\mathbb{R}}_{+}$ of a polynomial on ${\mathbb{R}}$, which has only real
roots.

###### Proof.

$1.\iff 2.$: This was observed in [2]. It is true even for fixed $A$ and $u$
and follows from the identities $|A+[0,u]|=|A|+|P_{u^{\bot}}A|$ and

$|\partial(A+[0,u])|=nV((A+[0,u])[n-1],B_{2}^{n})=|\partial
A|+|\partial(P_{u^{\bot}}A)|.$

$2.\implies 3.$: Define $T_{\varepsilon}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$
by $T_{\varepsilon}x=\varepsilon x+\langle x,v\rangle v$. Notice that
$T_{0}(B_{2}^{n})=[-v,v]$. From $(2)$ applied to $T_{\varepsilon}^{-1}A$ and
$T_{\varepsilon}^{-1}u$ we have

$|T_{\varepsilon}^{-1}A|V(T_{\varepsilon}^{-1}A[n-2],[0,T_{\varepsilon}^{-1}u],B_{2}^{n})\leq\frac{n}{n-1}V(T_{\varepsilon}^{-1}A[n-1],B_{2}^{n})V(T_{\varepsilon}^{-1}A[n-1],[0,T_{\varepsilon}^{-1}u]).$

Thus

$|A|V(A[n-2],[0,u],T_{\varepsilon}B_{2}^{n})\leq\frac{n}{n-1}V(A[n-1],T_{\varepsilon}B_{2}^{n})V(A[n-1],[0,u]).$

When $\varepsilon\to 0$, we get

$\displaystyle|A|V(A[n-2],[0,u],[0,v])\leq\frac{n}{n-1}V(A[n-1],[0,v])V(A[n-1],[0,u]).$
(27)

From (14) we get $|P_{u^{\bot}}A|_{n-1}=nV(A[n-1],[0,u])$ and

$|P_{[u,v]^{\bot}}A|_{n-2}\sqrt{1-\langle
u,v\rangle^{2}}=n(n-1)V(A[n-2],[0,u],[0,v]).$

$3.\iff 4.$: We may assume that $u$ is not colinear with $v$. Applying a
linear transformation to $A,u$ and $v$, we may assume that $u,v$ are
orthonormal. Expanding both sides of the inequality in 4. and using (9), we
get 3.
$3.\implies 5.$: As noticed above, $3.$ is equivalent to (27). From the
linearity of mixed volumes, we deduce that for every zonotopes $Z_{1}$ and
$Z_{2}$, one has

$\displaystyle|A|V(A[n-2],Z_{1},Z_{2})\leq\frac{n}{n-1}V(A[n-1],Z_{1})V(A[n-1],Z_{2}).$
(28)

Taking limits, we conclude that (28) is valid for every zonoids $Z_{1},Z_{2}$.
$5.\implies 2.$: Applying (28) to $Z_{1}=[0,u]$ and $Z_{2}=B_{2}^{n}$ and
using that
$V(A[n-2],[0,u],B_{2}^{n})=\frac{1}{n(n-1)}|\partial(P_{u^{\bot}}A)|$,
$V(A[n-1],[0,u])=\frac{1}{n}|P_{u^{\bot}}A|_{n-1}$ and
$V(A[n-1],B_{2}^{n})=\frac{1}{n}|\partial A|$, we conclude.
$3.\iff 6.$: We may assume that $u$ is not colinear with $v$. Applying a
linear transformation, to (3) and (6), it is enough to assume that $u,v$ are
orthonormal. Then

$P(t)=|A|+t(|P_{u^{\bot}}A|_{n-1}+|P_{v^{\bot}}A|_{n-1})+|P_{[u,v]^{\bot}}A|_{n-2}t^{2}.$

The equation

$|A|+t(|P_{u^{\bot}}A|_{n-1}+|P_{v^{\bot}}A|_{n-1})+|P_{[u,v]^{\bot}}A|_{n-2}t^{2}=0$

has real roots is equivalent to

$\displaystyle\left(\frac{|P_{u^{\bot}}A|_{n-1}+|P_{v^{\bot}}A|_{n-1}}{2}\right)^{2}\geq|A||P_{[u,v]^{\bot}}A|_{n-2},$
(29)

which follows immediately from 3.. To show that $6.\implies 3.$ assume (29) is
true for all $A$ in ${\cal L}$ and $u,v\in S^{n-1}$. Consider the linear
operator $T$ such that $Tu=tu$ and $Tv=t^{-1}v$ for $t>0$ and $Tx=x$ for
$x\in[u,v]^{\perp}$. Taking
$t=(|P_{u^{\bot}}A|_{n-1}/|P_{v^{\bot}}A|_{n-1}|)^{1/2}$ we get

$|P_{v^{\bot}}TA|_{n-1}=|P_{u^{\bot}}TA|_{n-1}.$

Applying (29) to $TA$, we get

$|P_{u^{\bot}}TA|_{n-1}|P_{v^{\bot}}TA|_{n-1}\geq|TA||P_{[u,v]^{\bot}}TA|_{n-2}.$

Since $|TA|=|A|$, $|P_{[u,v]^{\bot}}TA|_{n-2}=|P_{[u,v]^{\bot}}A|_{n-2}$ and

$|P_{u^{\bot}}TA|_{n-1}|P_{v^{\bot}}TA|_{n-1}=|P_{u^{\bot}}A|_{n-1}|P_{v^{\bot}}A|_{n-1}$

we get 3.  $\Box$

######  Remark 3.10.

It was proved in [45] that 5. in Theorem 3.9 is satisfied when $A$ is a
simplex (actually, even without constant $n/(n-1)$), thus all of the
properties in Theorem 3.9 are true for simplices. Actually, the inequalities
of statements 2. and 3. hold with an extra factor $\frac{n-1}{n}$ on the right
hand side.

######  Remark 3.11.

Notice that the inequality (24) shows that the property $5.$ of Theorem 3.9
holds true for the class $\mathcal{L}$ of compact convex sets in
${\mathbb{R}}^{2}$ and doesn’t hold in the class of compact convex sets in
${\mathbb{R}}^{n}$, for $n\geq 3$.

##  4 Some Special Cases

###  4.1 An improved inequality in ${\mathbb{R}}^{2}$

Inspired by an analogous result [15, Theorem 3] in Information Theory, T.
Courtade asked if

$|B|^{1/n}|C|^{1/n}+|A|^{1/n}|A+B+C|^{1/n}\leq|A+B|^{1/n}|A+C|^{1/n}$ (30)

for $A=B_{2}^{n}$ being the Euclidean ball, and any convex bodies $B,C$ in
${\mathbb{R}}^{n}$. Here we confirm Courtade’s conjecture in
${\mathbb{R}}^{2}$ in a more general setting.

######  Theorem 4.1.

Consider convex bodies $A,B,C\subset{\mathbb{R}}^{2}$, then

$|A|^{1/2}|A+B+C|^{1/2}+|B|^{1/2}|C|^{1/2}\leq|A+B|^{1/2}|A+C|^{1/2}.$ (31)

###### Proof.

The main tool to prove the above inequality is the following classical
inequality of Fenchel that we have already used. We will need now to use the
most general form of this inequality (see [44, (7.69) pp. 401]):

$(|A|V(B,C)-V(A,B)V(A,C))^{2}\leq(V(A,B)^{2}-|A||B|)(V(A,C)^{2}-|A||C|).$

Note that the above can be rewritten as

$|C|V(A,B)^{2}+|B|V(A,C)^{2}+|A|V^{2}(B,C)-|A||B||C|-2V(A,B)V(A,C)V(B,C)\leq
0$ (32)

Squaring both sides of (31) we get

$2\big{(}|A||A+B+C||B||C|\big{)}^{1/2}+|A||A+B+C|+|B||C|\leq|A+B||A+C|$

We use (9) to rewrite above as

$\left(|A||B||C||A+B+C|\right)^{1/2}+|A|V(B,C)\leq
2V(A,B)V(A,C)+V(A,B)|C|+|B|V(A,C).$

Using (24) we may rewrite the above inequality as

$|A||B||C||A+B+C|\leq\left(2V(A,B)V(A,C)+V(A,B)|C|+|B|V(A,C)-|A|V(B,C)\right)^{2}.$

Consider $rB$, $r\geq 0$, instead of $B$:

$|A||B||C||A+rB+C|\leq\left(2V(A,B)V(A,C)+V(A,B)|C|+r|B|V(A,C)-|A|V(B,C)\right)^{2}.$

The above represents a quadratic inequality $\alpha r^{2}+\beta r+\gamma\geq
0,$ with

$\alpha=|B|^{2}V^{2}(A,C)-|A||B|^{2}|C|,$
$\beta=2\big{(}2V(A,B)V(A,C)+V(A,B)|C|-|A|V(B,C)\big{)}|B|V(A,C)-2|A||B||C|V(B,A+C),$
$\gamma=\big{(}2V(A,B)V(A,C)+V(A,B)|C|-|A|V(B,C)\big{)}^{2}-|A||B||C||A+C|.$

It follows from (11) that $\alpha\geq 0$. It turns out $\beta$ may be negative
and thus we need to show that $D=\beta^{2}-4\alpha\gamma\leq 0,$ which after
division by $|B|^{2}$ becomes

$\left[\Big{(}2V(A,B)V(A,C)+V(A,B)|C|-|A|V(B,C)\Big{)}V(A,C)-|A||C|V(B,A+C)\right]^{2}$
$-\big{(}V^{2}(A,C)-|A||C|\big{)}\Big{[}\Big{(}2V(A,B)V(A,C)+V(A,B)|C|-|A|V(B,C)\Big{)}^{2}-|A||B||C||A+C|\Big{]}\leq
0.$

Simplifying the above inequality and dividing it by $|A||C|$ we may rearrange
the terms to get that our goal is to show that

$\Big{(}|C|V(A,B)^{2}-2V(A,B)V(A,C)V(B,C)+|A|V^{2}(B,C)\Big{)}\Big{(}|A|+2V(A,C)+|C|\Big{)}$
$+\big{(}V^{2}(A,C)-|A||C|\big{)}|B|(|A+C|)\leq 0.$

Factoring out $|A+C|$ we get that our goal is to show that

$\Big{(}|C|V(A,B)^{2}-2V(A,B)V(A,C)V(B,C)+|A|V^{2}(B,C)+|B|V^{2}(A,C)-|A||C||B|\Big{)}|A+C|\leq
0.$

Finally, the above inequality follows from (32).

$\Box$

From the failure of log-submodularity on the space of convex bodies for $n\geq
3$ (observed independently by Nayar and Tkocz [39] and a subset of the authors
[24]), we know that inequality (30) cannot possibly hold for $n\geq 3$ if $A$
is an arbitrary convex body. Of course, Courtade’s conjecture could still be
true since it only considers the case $A=B_{2}^{n}$. We note that a weaker
version of the conjecture, namely,

$|B_{2}^{n}||B_{2}^{n}+B+C|\leq|B_{2}^{n}+B||B_{2}^{n}+C|$

was proved in [24, Theorem 4.12] in the special case when $B$ is a zonoid and
$C$ is an arbitrary convex body.

###  4.2 Parallelotopes in general dimension

It is clear that 3. from Theorem 3.9 holds for ellipsoids. Indeed, after
applying a linear transformation, it reduces to the following inequality:
$|B_{2}^{n}||B_{2}^{n-2}|\leq|B_{2}^{n-1}|^{2}$, which follows from the log-
convexity of the Gamma function. In the next theorem, we prove that 3. from
Theorem 3.9 holds for the class of parallelotopes.

######  Theorem 4.2.

Let $A$ be a parallelotope in ${\mathbb{R}}^{n}$ and $u,v\in S^{n-1}$, then

$|A|\,|P_{[u,v]^{\bot}}A|_{n-2}\sqrt{1-\langle
u,v\rangle^{2}}\leq|P_{u^{\bot}}A|_{n-1}\,|P_{v^{\bot}}A|_{n-1},$

with equality if and only, when $A=\sum_{i=1}^{n}[a_{i},a_{i}+w_{i}],$ for
some $a_{i},w_{i}$ in ${\mathbb{R}}^{n}$ we have that $u=\sum_{i\in
I}\lambda_{i}w_{i}$ and $v=\sum_{i\in I^{c}}\lambda_{i}w_{i}$ for some
$I\in\\{1,\dots,n\\}$ and $\lambda_{1},\dots,\lambda_{n}$ in
${\mathbb{R}}^{n}$.

###### Proof.

We use the representation of the volume of projections using mixed volumes
(14) to restate the above statement as

$|A|V(A[n-2],[0,u],[0,v])\leq\frac{n}{n-1}V(A[n-1],[0,u])V(A[n-1],[0,v]).$

Applying a linear transformation, we may assume
$A=[0,1]^{n}=\sum_{i=1}^{n}[0,e_{i}]$. Thus

$V\left(A[n-1],[0,u]\right)=\frac{1}{n}\sum_{|I|=n-1}|{\mathop{\rm
det}}(u,(e_{i})_{i\in I})|=\frac{1}{n}\sum_{i=1}^{n}|u_{i}|,$
$V\left(A[n-2],[0,u],[0,v]\right)=\frac{1}{n(n-1)}\sum_{|I|=n-2}|{\mathop{\rm
det}}(u,v,(e_{i})_{i\in
I})|=\frac{1}{n(n-1)}\sum_{i<j}|u_{i}v_{j}-u_{j}v_{i}|.$

Finally we need to show

$\sum_{i<j}|u_{i}v_{j}-u_{j}v_{i}|\leq\sum|u_{i}|\sum|v_{j}|,$ (33)

which follows from the triangle inequality. The equality in (33) is only
possible if and only if $u_{i}v_{i}=0$ for every $i\in\\{1,\dots,n\\},$ which
implies the desired equality case.  $\Box$

We note that every zonotope $A$ can be seen as an orthogonal projection of a
high dimensional cube. Unfortunately, Theorem 4.2 can not be generalized
directly to the case of projection of higher co-dimensions (as it is done in
3. from Theorem 5.2 below), indeed such direct generalization requires Theorem
4.2 for all zonotopes in place of the cube. Thus we prove this property
directly for the case of parallelotopes in the next theorem.

######  Theorem 4.3.

Let $A$ be a parallelotope in ${\mathbb{R}}^{n}$, then

$|A||P_{E\cap F}A|\leq|P_{E}A||P_{F}A|.$ (34)

for any subspaces $E,F$ of ${\mathbb{R}}^{n}$ such that $E^{\perp}\subset F$.

###### Proof.

Notice that (34) is invariant under application of rotation $S\in O(n)$ to the
parallelotope $A$ and subspaces $E$ and $G.$ Thus, without loss of generality,
we may assume $A=T\left(\sum_{i=1}^{n}[0,e_{i}]\right),$ for some $T\in GL(n)$
and $E=\\{e_{1},\dots,e_{m}\\}^{\perp}$ and $F=\\{e_{m+1},\dots
e_{m+j}\\}^{\perp}$. Then

$|P_{E}\left(T\sum_{i=1}^{n}[0,e_{i}]\right)|_{n-m}=|\sum_{i=1}^{n}[0,P_{E}(Te_{i})]|_{n-m}=|\sum_{i=1}^{n}[0,P_{E}Te_{i}]+\sum_{k=1}^{m}[0,e_{k}]|_{n},$

where the last equality follows from
$\sum_{i=1}^{n}[0,P_{E}(Te_{i})]|_{n-m}\subset
E=\\{e_{1},\dots,e_{m}\\}^{\perp}$. Thus,

$\displaystyle|P_{E}\left(T\sum_{i=1}^{n}[0,e_{i}]\right)|_{n-m}=$
$\displaystyle\sum_{|I|=n-m}|{\mathop{\rm
det}}(e_{1},\dots,e_{m},\\{P_{E}Te_{i}\\}_{i\in I}|_{n}$ $\displaystyle=$
$\displaystyle\sum_{|I|=n-m}|{\mathop{\rm
det}}(e_{1},\dots,e_{m},\\{Te_{i}\\}_{i\in I}|_{n}$ $\displaystyle=$
$\displaystyle|{\mathop{\rm det}}(T)|\sum_{|I|=n-m}|{\mathop{\rm
det}}(T^{-1}e_{1},\dots,T^{-1}e_{m},\\{e_{i}\\}_{i\in I}|_{n}$
$\displaystyle=$ $\displaystyle|{\mathop{\rm
det}}(T)|\sum_{|J|=m}|{\mathop{\rm det}}(\\{w_{i}^{J}\\}_{i=1}^{m})|,$ (35)

where we denote by $w_{i}=T^{-1}e_{i},$ $i=1,\dots,n$ and by $u^{J}$ we denote
the orthogonal projection of vector $u$ onto the $span\\{e_{i}\\}_{i\in J}$.
We apply (4.2) to get that

$\displaystyle|P_{E\cap F}\sum_{i=1}^{n}[0,Te_{i}]|_{n-j-m}$ $\displaystyle=$
$\displaystyle|{\mathop{\rm det}}(T)|\sum_{|N|=m+j}|{\mathop{\rm
det}}(\\{w_{i}^{N}\\}_{i=1}^{m+j})|$ $\displaystyle=$
$\displaystyle|{\mathop{\rm
det}}(T)|\\!\\!\\!\sum_{|N|=m+j}\left|\sum_{|I|=m,I\subset
N}\\!\\!\\!\\!\\!\\!\varepsilon(N,I){\mathop{\rm det}}(\\{w_{i}^{I}\\}_{i\leq
m}){\mathop{\rm det}}(\\{w_{i}^{N\cap I^{c}}\\}_{i>m})\right|\\!,$

where in the last step we have used the Laplace formula. Finally,

$\displaystyle|P_{E\cap F}\sum_{i=1}^{n}[0,Te_{i}]|_{n-j-m}$
$\displaystyle\leq$ $\displaystyle|{\mathop{\rm
det}}(T)|\sum_{N=m+j}\sum_{|I|=m,I\subset N}|{\mathop{\rm
det}}(\\{w_{i}^{I}\\}_{i\leq m})||{\mathop{\rm det}}(\\{w_{i}^{N\cap
I^{c}}\\}_{i>m})|$ $\displaystyle=$ $\displaystyle|{\mathop{\rm
det}}(T)|\sum_{|I|=m}\sum_{N=m+j,I\subset J}|{\mathop{\rm
det}}(\\{w_{i}^{I}\\}_{i\leq m})||{\mathop{\rm det}}(\\{w_{i}^{N\cap
I^{c}}\\}_{i>m})|$ $\displaystyle=$ $\displaystyle|{\mathop{\rm
det}}(T)|\sum_{|I|=m}\sum_{I\cup J,|J|=j,I\cap J=\emptyset}|{\mathop{\rm
det}}(\\{w_{i}^{I}\\}_{i\leq m})||{\mathop{\rm det}}(\\{w_{i}^{J}\\}_{i>m})|$
$\displaystyle=$ $\displaystyle|{\mathop{\rm
det}}(T)|\sum_{|I|=m}|{\mathop{\rm det}}(\\{w_{i}^{I}\\}_{i\leq
m})|\sum_{|J|=j,}|{\mathop{\rm det}}(\\{w_{i}^{J}\\}_{i>m})|$ $\displaystyle=$
$\displaystyle|{\mathop{\rm
det}}(T)|^{-1}|P_{E}\left(T\sum_{i=1}^{n}[0,e_{i}]\right)|\,|P_{F}\left(T\sum_{i=1}^{n}[0,e_{i}]\right)|,$

where the last equality, again, follows from (4.2).  $\Box$

##  5 Inequalities for zonoids

###  5.1 Zonoids in ${\mathbb{R}}^{3}$

Zonoids form a natural class of bodies which is stable under addition and
linear transformations. In this section, we confirm property (4) from Theorem
3.9 in the class of three dimensional zonoids. Thus, using Theorem 5.1, we get
that all properties described in Theorems 3.6 and 3.9 are true for this class.

######  Theorem 5.1.

Let $A$ be a zonoid in ${\mathbb{R}}^{3}$ and $u,v\in S^{2}$. Then

$|A|_{3}|P_{[u,v]^{\bot}}A|_{1}\sqrt{1-\langle
u,v\rangle^{2}}\leq|P_{u^{\bot}}A|_{2}|P_{v^{\bot}}A|_{2}.$ (36)

###### Proof.

Using (15), inequality (36) is equivalent to

$|A|_{3}|V(A,[0,u],[0,v])\leq\frac{3}{2}V(A[2],[0,u])V(A[2],[0,v]).$ (37)

We assume that $u,v$ are linearly independent (otherwise the inequality is
trivial) and note that it is enough to prove (37) in the case of $u=e_{1}$,
$v=e_{2}$ and any zonoid $A$. Indeed, the more general case then follows by
applying the inequality to $T^{-1}A,$ where $T\in GL(3)$ is such that
$Te_{1}=u$ and $Te_{2}=v$. Thus our goal is to prove that, for any zonoid
$A\subset{\mathbb{R}}^{3}$,

$|A|_{3}|P_{[e_{1},e_{2}]^{\bot}}A|_{1}\leq|P_{e_{1}^{\bot}}A|_{2}|P_{e_{2}^{\bot}}A|_{2}.$
(38)

By approximation, it is enough to prove (38) when $A$ is a zonotope. Suppose
that $A=\sum_{i=1}^{M}[0,u_{i}],$ where
$u_{i}=(x_{i},y_{i},z_{i})\in{\mathbb{R}}^{3}$. Using (19), we get that (38)
is equivalent to

$\displaystyle\sum_{1\leq i<j<k\leq M}\left|{\mathop{\rm
det}}\\!\\!\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\ y_{i}&y_{j}&y_{k}\\\
z_{i}&z_{j}&z_{k}\end{pmatrix}\right|\sum_{i=1}^{M}|z_{i}|\leq\\!\\!\\!\sum_{1\leq
i<j\leq M}\left|{\mathop{\rm det}}\\!\\!\begin{pmatrix}y_{i}&y_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\sum_{1\leq i<j\leq M}\left|{\mathop{\rm
det}}\\!\\!\begin{pmatrix}x_{i}&x_{j}\\\ z_{i}&z_{j}\end{pmatrix}\right|$ (39)

We consider $y_{1},\dots,y_{M}$ and $z_{1},\dots,z_{M}$ as fixed, we write
$x=(x_{1},\dots,x_{M})\in{\mathbb{R}}^{M}$ and we define
$f,g:{\mathbb{R}}^{M}\to{\mathbb{R}}$ by

$f(x)=\sum_{1\leq i<j\leq M}\left|{\mathop{\rm
det}}\begin{pmatrix}x_{i}&x_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\quad\hbox{and}\quad g(x)=\sum_{1\leq i<j<k\leq
M}\left|{\mathop{\rm det}}\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\
y_{i}&y_{j}&y_{k}\\\ z_{i}&z_{j}&z_{k}\end{pmatrix}\right|.$

We note that $f$ and $g$ are piecewise affine and convex with respect to
$x_{i}$, for any $1\leq i\leq M$. We use the following elementary observation.
Let $\psi:{\mathbb{R}}\to{\mathbb{R}}$ be a convex function. Fix some positive
numbers $\\{a_{j}\\}_{j=1}^{K}$, $d\in{\mathbb{R}}$ and real numbers
$\\{c_{j}\\}_{j=1}^{K}$. To prove that, for all $t\in{\mathbb{R}}$,

$\psi(t)\leq\varphi(t):=\sum_{j=1}^{K}a_{i}|t+c_{i}|+d,$

it is enough to prove the inequality at all critical points $t=-c_{i}$ of
$\varphi$ and at the limit $t\to\pm\infty.$

We shall apply the above argument inductively to $f$ and $g$ as functions of
$x_{i}$, for $i\in\\{1,\dots,M\\}$, successively, with $x_{j}$ fixed for
$j\neq i$.

We start with $x_{1}$ and first check the limiting behavior at infinity (we
also prove the limit at infinity argument as a part of a more general
statement below). We note that, as $x_{1}\to\infty$, the left hand side of
(39) behaves like

$|x_{1}|\left(\sum_{1<j<k\leq M}\left|{\mathop{\rm
det}}\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}\right|\right)\left(\sum_{j=1}^{M}|z_{j}|\right)$

and the right hand side of (39) behaves like

$|x_{1}|\left(\sum_{1\leq j<k\leq M}\left|{\mathop{\rm
det}}\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}\right|\right)\left(\sum_{j=2}^{M}|z_{j}|\right).$

Thus, (39) becomes

$\left(\sum_{1<j<k\leq M}\left|{\mathop{\rm det}}\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}\right|\right)\left(\sum_{j=1}^{M}|z_{j}|\right)\leq\left(\sum_{1\leq
j<k\leq M}\left|{\mathop{\rm det}}\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}\right|\right)\left(\sum_{j=2}^{M}|z_{j}|\right)$

or

$\left(\sum_{1<j<k\leq M}\left|{\mathop{\rm det}}\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}\right|\right)|z_{1}|\leq\left(\sum_{1<k\leq
M}\left|{\mathop{\rm det}}\begin{pmatrix}y_{1}&y_{k}\\\
z_{1}&z_{k}\end{pmatrix}\right|\right)\left(\sum_{j=2}^{M}|z_{j}|\right).$

The above equation is exactly the ${\mathbb{R}}^{2}$ analog of (36), with
$A=\sum_{i=2}^{M}[0,(y_{i},z_{i})]$; $u=[1,0]$ and
$v=(y_{1},z_{1})/|(y_{1},z_{1})|$, thus it holds.

Our next goal is to study the critical points of $f$ with respect to $x_{1}$,
which satisfy

${\mathop{\rm det}}\begin{pmatrix}x_{1}&x_{j}\\\
z_{1}&z_{j}\end{pmatrix}=0,\mbox{ for }z_{j}\not=0,\,\,\,2\leq j\leq M.$

When $x_{1}$ is a solution of the above equation, then $(x_{j},z_{j})$ is
parallel to $(x_{1},z_{1})$. Assume (without loss of generality) that $j=2$
and $(x_{1},z_{1})=\lambda(x_{2},z_{2}).$ We study (39), with respect to
$x_{2}$, under the assumption that $(x_{1},z_{1})=\lambda(x_{2},z_{2})$ and
continue this algorithm inductively (we will show the general step below).

We must also consider the case when $z_{j}=0$, for all $j\geq 2$. In this case
(39) becomes

$\left(|z_{1}|\sum_{1<j<k\leq M}\left|{\mathop{\rm
det}}\begin{pmatrix}x_{j}&x_{k}\\\
y_{j}&y_{k}\end{pmatrix}\right|\right)|z_{1}|\leq\left(|z_{1}|\sum_{j=2}^{M}|y_{j}|\right)\left(|z_{1}|\sum_{j=2}^{M}|x_{j}|\right)$

or

$\left(\sum_{1<j<k\leq M}\left|{\mathop{\rm det}}\begin{pmatrix}x_{j}&x_{k}\\\
y_{j}&y_{k}\end{pmatrix}\right|\right)\leq\left(\sum_{j=2}^{M}|y_{j}|\right)\left(\sum_{j=2}^{M}|x_{j}|\right),$

which follows immediately from

$\left|{\mathop{\rm det}}\begin{pmatrix}x_{j}&x_{k}\\\
y_{j}&y_{k}\end{pmatrix}\right|\leq|x_{j}y_{k}|+|x_{k}y_{j}|.$ (40)

Continuing this process, we arrive to the case

$(x_{1},z_{1})=\lambda_{1}(x_{m},z_{m}),\dots,(x_{m-1},z_{m-1})=\lambda_{m-1}(x_{m},z_{m}),$
(41)

for some $2\leq m\leq M$. We also denote $\lambda_{m}=1$ and we study $f$ and
$g$ as functions of $x_{m}$.

Again, our first step is to confirm (39), when $x_{m}\to\pm\infty.$ To do so,
let us see how the functions $f$ and $g$ changed under (41). Let us first
consider the terms appearing in $g$:

$\left|{\mathop{\rm det}}\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\
y_{i}&y_{j}&y_{k}\\\ z_{i}&z_{j}&z_{k}\end{pmatrix}\right|,$

when $i<j<k$. For $m<i$, the above determinant doesn’t depend on $x_{m}$, so
we only consider the case when $m\geq i$.

  * •

If $m\geq k$, then the determinant is zero.

  * •

if $i\leq m<j$ then, when $|x_{m}|\to\infty$

$\left|{\mathop{\rm det}}\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\
y_{i}&y_{j}&y_{k}\\\
z_{i}&z_{j}&z_{k}\end{pmatrix}\right|\sim|x_{i}|\left|{\mathop{\rm
det}}\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}\right|=|\lambda_{i}||x_{m}|\left|{\mathop{\rm
det}}\begin{pmatrix}y_{j}&y_{k}\\\ z_{j}&z_{k}\end{pmatrix}\right|$

  * •

$j\leq m<k$ then, when $|x_{m}|\to\infty,$ observing that
$\lambda_{i}=z_{i}/z_{m}$ and $\lambda_{j}=z_{j}/z_{m}$ we get

$\left|{\mathop{\rm det}}\\!\\!\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\
y_{i}&y_{j}&y_{k}\\\
z_{i}&z_{j}&z_{k}\end{pmatrix}\\!\\!\right|\\!\sim\\!\left|\lambda_{i}x_{m}{\mathop{\rm
det}}\\!\\!\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}-\lambda_{j}x_{m}{\mathop{\rm
det}}\\!\\!\begin{pmatrix}y_{i}&y_{k}\\\
z_{i}&z_{k}\end{pmatrix}\right|=|x_{m}|\frac{|z_{k}|}{|z_{m}|}\left|{\mathop{\rm
det}}\\!\\!\begin{pmatrix}y_{i}&y_{j}\\\ z_{i}&z_{j}\end{pmatrix}\right|.$

We also need to compute the behaviour of the terms appearing in $f$:

$\left|{\mathop{\rm det}}\begin{pmatrix}x_{i}&x_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|,$

for $i<j$. When $m\geq j$, it is zero and when $m<i$, it does not dependent on
$x_{m}$. So we assume that $i\leq m<j$. We get

$\left|{\mathop{\rm det}}\begin{pmatrix}x_{i}&x_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\sim|\lambda_{i}|\,|x_{m}|\,|z_{j}|.$

Thus to show that the (39) is true, as $|x_{m}|\to\infty$, we need to prove
that

$\left[(\sum_{i\leq m}|\lambda_{i}|)\sum_{m<j<k<M}\left|{\mathop{\rm
det}}\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}\right|+\frac{1}{|z_{m}|}\left(\sum_{m<k}|z_{k}|\right)\\!\\!\\!\left(\sum_{1\leq
i<j\leq m}\left|{\mathop{\rm det}}\begin{pmatrix}y_{i}&y_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\right)\right]\\!\\!\\!\left(\sum_{i=1}^{M}|z_{i}|\right)$
$\leq\left(\sum_{1\leq i<j\leq M}\left|{\mathop{\rm
det}}\begin{pmatrix}y_{i}&y_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\right)\left(\sum_{i\leq
m}|\lambda_{i}|\right)\sum_{j>k}|z_{k}|.$

Multiplying both sides by $|z_{m}|$, we are reduced to

$\left[(\sum_{i\leq m}|z_{i}|)\sum_{m<j<k<M}\left|{\mathop{\rm
det}}\begin{pmatrix}y_{j}&y_{k}\\\
z_{j}&z_{k}\end{pmatrix}\right|+\left(\sum_{m<k}|z_{k}|\right)\left(\sum_{1\leq
i<j\leq m}\left|{\mathop{\rm det}}\begin{pmatrix}y_{i}&y_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\right)\right]\left(\sum_{i=1}^{M}|z_{i}|\right)$
$\leq\left(\sum_{1\leq i<j\leq M}\left|{\mathop{\rm
det}}\begin{pmatrix}y_{i}&y_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\right)\left(\sum_{i\leq
m}|z_{i}|\right)\sum_{j>k}|z_{k}|$

Let $A^{\prime}=\sum_{i=m}^{M}[0,(y_{i},z_{i})]$ and
$B^{\prime}=\sum_{i=1}^{m}[0,(y_{i},z_{i})]$, then the above becomes:

$\left[|P_{e_{1}^{\perp}}B^{\prime}||A^{\prime}|+|P_{e_{1}^{\perp}}A^{\prime}||B^{\prime}|\right]|P_{e_{1}^{\perp}}(A^{\prime}+B^{\prime})|\leq|A^{\prime}+B^{\prime}||P_{e_{1}^{\perp}}B^{\prime}||P_{e_{1}^{\perp}}A^{\prime}|.$

This is Bonnesen’s inequality (6) in ${\mathbb{R}}^{2}$

$\frac{|A^{\prime}|_{2}}{|P_{e_{1}^{\bot}}A|_{1}}+\frac{|B^{\prime}|_{2}}{|P_{e_{1}^{\bot}}B^{\prime}|_{1}}\leq\frac{|A^{\prime}+B^{\prime}|_{2}}{|P_{e_{1}^{\bot}}(A^{\prime}+B^{\prime})|_{1}}.$

Our next step (if $m<M$) is to consider the critical points of $f$, as a
function of $x_{m}$; they are given by the equations

${\mathop{\rm det}}\begin{pmatrix}x_{m}&x_{j}\\\ z_{m}&z_{j}\end{pmatrix}=0,$

for $z_{j}\not=0,$ $j\geq m+1$, if such $z_{j}$ exists and repeat the process
for all $m<M$. If $z_{j}=0,$ for all $j\geq m+1$, let us confirm the
inequality directly (as for the case $m=1$). To calculate

$\left|{\mathop{\rm det}}\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\
y_{i}&y_{j}&y_{k}\\\ z_{i}&z_{j}&z_{k}\end{pmatrix}\right|$

we may consider cases: if $i,j,k\leq m$, then the rank of the above matrix is
at most to $2$ and the determinant is zero; if $i,j,k\leq m$, the matrix has a
row of zeros and the determinant is again zero. In the case when $i\leq
m<j<k$, we get

$\left|{\mathop{\rm det}}\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\
y_{i}&y_{j}&y_{k}\\\
z_{i}&z_{j}&z_{k}\end{pmatrix}\right|=|z_{i}|\left|{\mathop{\rm
det}}\begin{pmatrix}x_{j}&x_{k}\\\ y_{j}&y_{k}\end{pmatrix}\right|.$

When $i<j\leq m<k$, we use that $(x_{i},x_{j})$ is parallel to $(z_{i},z_{j})$
to get that

$\left|{\mathop{\rm det}}\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\
y_{i}&y_{j}&y_{k}\\\
z_{i}&z_{j}&z_{k}\end{pmatrix}\right|=|x_{k}|\left|{\mathop{\rm
det}}\begin{pmatrix}y_{i}&y_{j}\\\ z_{i}&z_{j}\end{pmatrix}\right|.$

We make a similar analysis on the right hand side of (39) which becomes

$\displaystyle\left(\sum_{i=1}^{m}|z_{i}|\right)\left[\left(\sum_{i=1}^{m}|z_{i}|\right)\sum_{m<j<k}\left|{\mathop{\rm
det}}\begin{pmatrix}x_{j}&x_{k}\\\
y_{j}&y_{k}\end{pmatrix}\right|+\sum\limits_{k>m}|x_{k}|\sum_{i<j\leq
m}\left|{\mathop{\rm det}}\begin{pmatrix}y_{i}&y_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\right]$
$\displaystyle\leq\left[\left(\sum_{i=1}^{m}|z_{i}|\right)\left(\sum_{j>m}|y_{j}|\right)+\sum_{i<j\leq
m}\left|{\mathop{\rm det}}\begin{pmatrix}y_{i}&y_{j}\\\
z_{i}&z_{j}\end{pmatrix}\right|\right]\left[\left(\sum_{i=1}^{m}|z_{i}|\right)\left(\sum_{j>m}|x_{j}|\right)\right]$

The above inequality follows directly by simplification and application of
(40).

We repeat the above process until we have $m=M$, thus
$(x_{i},z_{i})=\lambda_{i}(x_{M},z_{M})$ for each $i=1,\dots,M-1$. Thus, the
left hand side of the inequality (39) is equal to zero. Indeed, each of the
following matrices

$\begin{pmatrix}x_{i}&x_{j}&x_{k}\\\ y_{i}&y_{j}&y_{k}\\\
z_{i}&z_{j}&z_{k}\end{pmatrix}$

has rank less or equal then $2$.  $\Box$

###  5.2 More equivalent formulations for zonoids

In this section, we show some additional equivalences (which thus hold in
dimension $3$).

######  Theorem 5.2.

Let $n\in{\mathbb{N}}$, then the following are equivalent.

  1. 1.

For every $k\geq n$ and for every family of vectors $u_{1},\dots,u_{k}$ in
${\mathbb{R}}^{n}$ the function $f:2^{[k]}\to{\mathbb{R}}$ defined, for
$S\subset[k]$, by

$f(S)=\log\left|\sum_{i\in S}[0,u_{i}]\right|=\log\sum_{I\subset
S,|I|=n}|{\mathop{\rm det}}(\\{u_{i}\\}_{i\in I})|$

is submodular: for all zonoids $A,B,C$ one has

$|A||A+B+C|\leq|A+B||A+C|.$

  2. 2.

For every $k\geq n$ and for every family of vectors $u_{1}\dots,u_{k}$ in
${\mathbb{R}}^{n}$, for every $u,v\in{\mathbb{R}}^{n}$

$\displaystyle\sum_{|I|=n}|{\mathop{\rm det}}(\\{u_{i}\\}_{i\in I})|$
$\displaystyle\sum_{|I|=n-2}|{\mathop{\rm det}}(u,v,(u_{i})_{i\in I})|$
$\displaystyle\leq\sum_{|I|=n-1}|{\mathop{\rm det}}(u,(u_{i})_{i\in
I})|\sum_{|I|=n-1}|{\mathop{\rm det}}(v,(u_{i})_{i\in I})|.$

  3. 3.

For every zonoid $A$ in ${\mathbb{R}}^{n}$ and every subspaces $E,F$ of
${\mathbb{R}}^{n}$ such that $E^{\perp}\subseteq F$ we have

$|A||P_{E\cap F}A|\leq|P_{E}A||P_{F}A|.$

  4. 4.

For every $m=1,\dots,n$, for every zonoid $A$ and every orthonormal sequence
$u_{1},\dots,u_{m}$, one has

$|A|^{m-1}\left|P_{[u_{1},\dots,u_{m}]^{\bot}}A\right|_{n-m}\leq\prod_{i=1}^{m}\left|P_{u_{i}^{\bot}}A\right|_{n-1}.$

  5. 5.

For every $m=1,\dots,n$ and all zonoids $A,B_{1},\dots,B_{m}$ in
${\mathbb{R}}^{n}$, one has

$|A|^{m-1}V(A[n-m],B_{1},\dots,B_{m})\leq\frac{n^{m}(n-m)!}{n!}\prod_{i=1}^{m}V(A[n-1],B_{i}).$

###### Proof.

The proof is based on translations of the properties described in Theorems 3.6
and 3.9 to the properties of zonoids.

We first show that 1. in Theorem 5.2 is equivalent to 1. in Theorem 3.6 in the
class of zonotopes. Indeed, $f$ is submodular if and only if for every
$S,T\subset\\{1,\dots,k\\}$

$\left|\sum_{i\in S\cup T}[0,u_{i}]\right|\left|\sum_{i\in S\cap
T}[0,u_{i}]\right|\leq\left|\sum_{i\in S}[0,u_{i}]\right|\left|\sum_{i\in
T}[0,u_{i}]\right|,$

which is 1. in Theorem 5.2 for zonotopes $A=\sum_{i\in S\cap T}[0,u_{i}]$,
$B_{1}=\sum_{i\in S\setminus T}[0,u_{i}]$ and $B_{2}=\sum_{i\in T\setminus
S}[0,u_{i}]$. We use (19) to finish the proof.

We next note that 2. in Theorem 5.2 is equivalent to 3. of Theorem 3.9 in the
case of zonotopes. Assume by homogeneity that $u,v\in S^{n-1}$. Then we apply
(19) to get:

$\left|\sum_{i=1}^{m}[0,u_{i}]\right|=\sum_{I\subset[m],|I|=n}|{\mathop{\rm
det}}(u_{i})_{i\in I}|\mbox{ and
}\left|P_{v^{\perp}}\left(\sum_{i=1}^{m}[0,u_{i}]\right)\right|=\sum_{I\subset[m],|I|=n-1}|{\mathop{\rm
det}}(v,(u_{i})_{i\in I})|$

and the similar formula for the volume of
$P_{[u,v]^{\perp}}\left(\sum_{i=1}^{m}[0,u_{i}]\right)$.

Next, we show that 3. is equivalent to 3. from Theorem 3.9, which can be
restated as

$\displaystyle|A||P_{[e_{i},e_{j}]^{\bot}}A|_{n-2}\leq|P_{e_{i}^{\bot}}A|_{n-1}|P_{e_{j}^{\bot}}A|_{n-1},$
(42)

for any zonoid $A$ and $i\not=j$, where $(e_{1},\dots,e_{n})$ is any
orthonormal basis. Moreover 3. from Theorem 5.2 is equivalent to

$\displaystyle|A||P_{[e_{1},\dots
e_{k}]^{\perp}}A|_{n-k}\leq|P_{[e_{1},\dots,e_{i}]^{\perp}}A|_{n-i}|P_{[e_{i+1},\dots,e_{k}]^{\perp}}A|_{n-(k-i)}.$
(43)

for any $i<k\leq n$ and any zonoid $A$ in ${\mathbb{R}}^{n}$. Thus (42) is a
particular case of (43). To prove the reverse, we first notice that if
$(\ref{eq:z2})$ holds for any zonoid $A$ in ${\mathbb{R}}^{n}$, then it also
must hold for zonoids in any dimension $m\leq n$. Indeed, for any zonoid $A$
in ${\mathbb{R}}^{m}$, the cylinder $A\times[0,1]^{n-m}$ is a zonoid. Next, we
may prove property (43) by induction. Indeed, using (42), it is true for $k=2$
and $i=1$, any $n\in\mathbb{N}$ and any zonoid $A$ in ${\mathbb{R}}^{n}$.
Assume the statement is true for some $k\in\mathbb{N}$ any $i<k\leq n$. Let us
apply the statement to the zonoid $P_{e_{k+1}^{\perp}}A$ to get

$\displaystyle|P_{e_{k+1}^{\perp}}A|_{n-1}|P_{[e_{1},\dots,e_{k},e_{k+1}]^{\perp}}$
$\displaystyle A|_{n-(k+1)}$ (44)
$\displaystyle\leq|P_{[e_{1},\dots,e_{i},e_{k+1}]^{\perp}}A|_{n-(i+1)}|P_{[e_{i+1},\dots,e_{k},e_{k+1}]^{\perp}}A|_{n-(k+1-i)}.$

In addition, we apply the inductive hypothesis to $A$ and the subspace spanned
by $\\{e_{1},\dots,e_{i},e_{k+1}\\}$ and the subspace spanned by $e_{k+1}$ to
get

$\displaystyle|A||P_{[e_{1},\dots,e_{i},e_{k+1}]^{\perp}}A|_{n-(i+1)}\leq|P_{[e_{1},\dots,e_{i}]^{\perp}}A|_{n-i}|P_{e_{k+1}^{\perp}}A|_{n-1}.$
(45)

Finally, we multiply (44) and (45) to finish the proof.

To prove that 3. implies 4., we apply (43) to $k=m$ and $i=1$, we get that

$\frac{|P_{[u_{1},\dots,u_{m}]^{\perp}}A|}{|P_{[u_{2},\dots,u_{m}]^{\perp}}A|}\leq\frac{|P_{u_{1}^{\perp}}A|}{|A|}.$

In the same way, for every $1\leq i\leq m$, one has

$\frac{|P_{[u_{i},\dots,u_{m}]^{\perp}}A|}{|P_{[u_{i+1},\dots,u_{m}]^{\perp}}A|}\leq\frac{|P_{u_{i}^{\perp}}A|}{|A|}.$

Taking the product of these inequalities, we get the result.

To prove that 4. implies 5., we use (15). Thus (4) gives that for every
orthonormal family of vectors $u_{1},\dots,u_{m}$, one has

$|A|^{m-1}V(A[n-m],[0,u_{1}],\dots,[0,u_{m}])\leq\frac{n^{m}(n-m)!}{n!}\prod_{i=1}^{m}V(A[n-1],[0,u_{i}]).$

Since this inequality is invariant with respect to any linear image of $A$ by
an invertible map, it holds also for any independent $u_{1},\dots,u_{m}$.
Then, we deduce that 5. holds for any sums of segments. The inequality for
zonoids follows by taking limits.

Finally, 5. with $m=2$ is equivalent to 5. in Theorem 3.9.  $\Box$

Notice that 5. of Theorem 5.2 can be rephrased by saying that, for any zonoid
$A$ in ${\mathbb{R}}^{n}$, the function $f:[n]\to{\mathbb{R}}$ defined by
$f(S)=\log(|P_{[e_{i};i\in S]}A|)$ is submodular.

The inequalities analogous to 3. in Theorems 3.9 and 5.2 belong to the class
of local Loomis-Whitney type inequalities and were studied in many works,
including [27, 22, 45, 2, 1], for the most general classes of convex bodies.
In the next lemma, we present a new proof of a result from [27], the proof
uses the approach of the proof of Theorem 3.9.

######  Lemma 5.3.

Consider a convex body $K$ in ${\mathbb{R}}^{n}$ and a pair of orthogonal
vectors $u,v\in S^{n-1}$, then

$|K||P_{[u,v]^{\perp}}K|\leq\frac{2(n-1)}{n}|P_{u^{\perp}}K||P_{v^{\perp}}K|.$

###### Proof.

Let $L=[0,u]+\alpha[0,v],$ where $\alpha=|P_{u^{\perp}}K|/|P_{v^{\perp}}K|,$
noticing that the case $|P_{v^{\perp}}K|=0$ is trivial. Then

$nV(K[n-1],L)=|P_{u^{\perp}}K|_{n-1}+\alpha|P_{v^{\perp}}K|_{n-1}\mbox{ and
}\frac{n(n-1)}{2}V(K[n-2],L[2])=\alpha|P_{[u,v]^{\perp}}K|.$

Using Minkowski’s second inequality (12), we get

$\left(\frac{|P_{u^{\perp}}K|+\alpha|P_{v^{\perp}}K|}{n}\right)^{2}\geq\frac{2\alpha}{n(n-1)}|K||P_{[u,v]^{\perp}}K|$

we substitute $\alpha=|P_{u^{\perp}}K|/|P_{v^{\perp}}K|$ to finish the proof.
$\Box$

The main tool in the proof of Lemma 5.3 is Minkowski’s second inequality,
which relies on the fact that the polynomial $Q(t)=|K+tL|$ raised to the power
$1/n$ is a concave function for $t\geq 0$. We conjecture that the concavity
properties of this polynomial can be improved for $Z$ being a zonoid and $L$
being a finite sum of $m\leq n$ segments. More precisely, we conjecture that
for any $m\leq n$ and any sequence of vectors $u_{1},\dots,u_{m}$, $m\leq n$
from ${\mathbb{R}}^{n}$ if $P(t)=|Z+t\sum_{i=1}^{m}[0,u_{i}]|,$ then
$P^{1/m}(t)$ is a concave function for $t\geq 0$. This conjecture would follow
if the statements of Theorem 5.2 or Theorem 3.9 would be true. Still we can
prove the following proposition in ${\mathbb{R}}^{3}$.

######  Proposition 5.4.

Let $Z$ be a zonoid in ${\mathbb{R}}^{3}$ and $u,v$ be two vectors from
${\mathbb{R}}^{3}$. Let $P(t)=|Z+t([0,u]+[0,v])|,$ then $P^{1/2}(t)$ is a
concave function for $t\geq 0$.

###### Proof.

We may assume that the vectors $u,v$ are linearly independent. Then, applying
a linear transformation $T$ to $Z$ and $u,v$ we may assume that vectors $u,v$
are orthogonal to each others and belong to $S^{2}.$ Using that
$Z+t([0,u]+[0,v])$ is again a zonoid, it is enough for us to show that
$(P^{1/2}(0))^{\prime\prime}\leq 0$, or $2P(0)P^{\prime\prime}(0)\leq
P^{\prime}(0)^{2}$. Using that

$P(0)=|Z|,\quad
P^{\prime}(0)=|P_{u^{\perp}}Z|+|P_{v^{\perp}}Z|\quad\hbox{and}\quad
P^{\prime\prime}(0)=2|P_{[u,v]^{\bot}}Z|.$

and Theorem 5.1, we get

$2P(0)P^{\prime\prime}(0)=4|Z||P_{[u,v]^{\bot}}Z|\leq
4|P_{u^{\bot}}Z||P_{v^{\bot}}Z|\leq(|P_{u^{\perp}}K|+|P_{v^{\perp}}K|)^{2}=P^{\prime}(0)^{2}.$

$\Box$

It follows from part 6. of Theorem 3.9 and Theorem 5.1 that if
$P(t)=|Z+t([0,u]+[0,v])|$, then $P(t)$, as a polynomial on ${\mathbb{R}}$, has
only real roots, for any zonoid $Z$ and any $u,v\in{\mathbb{R}}^{3}$. Thus
another way to prove Proposition 5.4 is to notice the following simple
property. Consider a quadratic polynomial $P$ with positive coefficients, then
$\sqrt{P(t)}$ is concave for $t\geq 0$ if and only if $P$ has only real roots.
Thus, to study concavity of $\sqrt{P_{C}(t)}=|Z+tC|^{1/2},$ when $C$ is two
dimensional, one may study the roots of $P$. The subject of roots of Steiner-
type polynomials has attracted a fair bit of attention in the literature, see,
e.g., [29] and references therein. Proposition 5.4 makes the following
conjecture plausible:

######  Question 5.5.

Let $Z$ be a zonoid in ${\mathbb{R}}^{n}$. Then is it true that the polynomial
$P_{C}(t)=|Z+tC|$ has only real roots for every convex body $C$ of dimension
$2$?

This question is directly connected to a question of Adam Marcus that we
learned from Guillaume Aubrun about the roots of Steiner polynomials of
zonoids, and, in fact, it was one of the starting point of this investigation.

######  Question 5.6.

Let $Z$ be a zonoid in ${\mathbb{R}}^{n}$. Then is it true that the Steiner
polynomial $P_{Z}(t)=|Z+tB_{2}^{n}|$ has only real roots?

Observe that, in the plane, even more is true: for any convex bodies $K,L$,
the polynomial $P_{K,L}(t)=|K+tL|$ has only real roots. Indeed,
$\sqrt{P_{K,L}(t)}$ is concave for $t\geq 0$ by Brunn-Minkowski inequality.
This can be also seen from the computing the discriminant and noticing that
$V(K,L)^{2}-|K||L|\geq 0$.

Analogously, Question 5.5 is equivalent to the following question for mixed
volumes: fix $n\geq 3$, and let $K$ a zonoid in ${\mathbb{R}}^{n}$ and $L$ be
a two dimensional zonoid, is it true that

$|K|V(K[n-2],L[2])\leq\frac{n}{2(n-1)}V(K[n-1],L)^{2}?$

The above inequality is true for $L$ being a parallelogram, as follows from
Theorem 3.6. But, it is not true for general zonoids, as we show in the
following proposition inspired by a work by Victor Katsnelson [30].

######  Proposition 5.7.

Let $n\geq 3$. Then there exists a zonoid $Z$ in ${\mathbb{R}}^{n}$ such that
the Steiner polynomial $P_{Z}(t)=|Z+tB_{2}^{n}|$ and the polynomial
$Q_{Z}(t)=|tZ+B_{2}^{n}|$ has roots which are not real.

###### Proof.

Noticing that $Q_{Z}(t)=t^{n}P_{Z}(1/t),$ it is enough to show that $P_{Z}(t)$
has a non-real root. Consider
$Z=B_{2}^{2}\times\\{0\\}\subset{\mathbb{R}}^{2}\times{\mathbb{R}}^{n-2}$.
Integrating on sections parallel to ${\mathbb{R}}^{2}\times\\{0\\}$ and
changing variable, we get that, for $t\geq 0$,

$P_{Z}(t)=|Z+tB_{2}^{n}|=|B_{2}^{2}|\int_{tB_{2}^{n-2}}\left(1+\sqrt{t^{2}-|x|^{2}}\right)^{2}dx=t^{n-2}|B_{2}^{2}|\int_{B_{2}^{n-2}}\left(1+t\sqrt{1-|x|^{2}}\right)^{2}dx.$

Since this last expression is the restriction of a polynomial to $t\geq 0$,
the equality between $P_{Z}(t)$ and the last term is valid on whole
${\mathbb{R}}$. Let $t\in{\mathbb{R}}$ be fixed. For almost all $x\in
B_{2}^{n-2}$, $(1+t\sqrt{1-|x|^{2}})^{2}>0$, thus $P_{Z}(t)>0$, for $t\neq 0$,
so the only real root of $P_{Z}$ is $0$ and it is of order $n-2$. Hence
$P_{Z}$ has exactly $2$ non-real roots.  $\Box$

Proposition 5.7 shows that the answers to both Question 5.5 and Question 5.6
are negative.

Note that the example of the zonoid $Z$ created in Proposition 5.7 is flat
(i.e., has an affine hull of lower dimension than the ambient space), but one
can replace it by a non-flat zonoid by using a small perturbation and the
continuity of the roots of polynomials.

##  6 Inequalities for $L_{p}$-zonoids

Firey [20] extended the concept of Minkowski sum and introduced, for each real
$p\geq 1$, a $p$-linear combination of convex bodies, the so-called
$\ell_{p}$-sum $K\oplus_{p}L$ of the convex bodies $K$ and $L$ containing the
origin by:

$h_{K\oplus_{p}L}(x)=\left(h_{K}(x)^{p}+h_{L}(x)^{p}\right)^{\frac{1}{p}},\quad\forall
x\in{\mathbb{R}}^{n}.$

For any linear transform $T$ on ${\mathbb{R}}^{n}$, one has
$T(K\oplus_{p}L)=(TK)\oplus_{p}(TL)$. In a series of papers, Lutwak [31, 32]
showed that the Firey sums lead to a Brunn-Minkowski theory for each $p\geq
1$, including $L_{p}$-Brunn-Minkowski inequality, definition and inequalities
for $L_{p}$-mixed volumes, $L_{p}$-Minkowski problem, as well as many other
applications. In this section, we show the connection of the discussion from
previous sections to this theory.

An $L_{p}$-zonotope is the $\ell_{p}$-sum of centered segments and an
$L_{p}$-zonoid is the Hausdorff limit of $L_{p}$-zonotopes. For $p=2$, an
$L_{2}$-zonoid is always an ellipsoid, possibly living in a lower-dimensional
subspace, thus it can be written as the sum of $m\leq n$ orthogonal segments
and is therefore an $L_{2}$-zonotope. The following extension of Conjecture 5
is thus natural.

######  Question 6.1.

Let $p\geq 1$ and consider $L_{p}$-zonoids $A,B$ in ${\mathbb{R}}^{n}$ is it
true that

$\left(\frac{|A\oplus_{p}B|}{|P_{u^{\bot}}(A\oplus_{p}B)|_{n-1}}\right)^{p}\geq\left(\frac{|A|}{|P_{u^{\bot}}A|_{n-1}}\right)^{p}+\left(\frac{|B|}{|P_{u^{\bot}}B|_{n-1}}\right)^{p}?$
(46)

###  6.1 The case $p=2$

The next theorem gives an affirmative answer to this question in the case
$p=2$.

######  Theorem 6.2.

Let $A,B$ be a pair of $L_{2}$-zonoids in ${\mathbb{R}}^{n}$ and let $u$ in
$S^{n-1}$. Then

$\left(\frac{|A\oplus_{2}B|}{|P_{u^{\bot}}(A\oplus_{2}B)|_{n-1}}\right)^{2}\geq\left(\frac{|A|}{|P_{u^{\bot}}A|_{n-1}}\right)^{2}+\left(\frac{|B|}{|P_{u^{\bot}}B|_{n-1}}\right)^{2},$

with equality if and only if $A$ and $B$ have parallel tangent hyperplanes at
$\rho_{A}(u)u$ and $\rho_{B}(u)u$.

We first give two proofs of the inequality and then prove the equality case.

Proof 1: For $m\geq n$ and $u_{1},\dots,u_{m}\in{\mathbb{R}}^{n}$, let $U$ be
the $n\times m$ matrix whose columns are $u_{1},\dots,u_{m}$. One has
$[-u_{1},u_{1}]\oplus_{2}\cdots\oplus_{2}[-u_{m},u_{m}]=UB_{2}^{m}=\sqrt{UU^{*}}B_{2}^{n}$.
Indeed,

$\displaystyle
h^{2}_{[-u_{1},u_{1}]\oplus_{2}\cdots\oplus_{2}[-u_{m},u_{m}]}(x)$
$\displaystyle=\sum_{i=1}^{m}\langle u_{i},x\rangle^{2}=\sum_{i=1}^{m}\langle
Ue_{i},x\rangle^{2}=\sum_{i=1}^{m}\langle
e_{i},U^{*}x\rangle^{2}=|U^{*}x|^{2}=h^{2}_{UB_{2}^{m}}(x)$
$\displaystyle=\langle
U^{*}x,U^{*}x\rangle=\langle\sqrt{UU^{*}}x,\sqrt{UU^{*}}x\rangle=|\sqrt{UU^{*}}x|^{2}=h^{2}_{\sqrt{UU^{*}}B_{2}^{n}}(x).$

Using this and the Cauchy-Binet formula, one has

$|[-u_{1},u_{1}]\oplus_{2}\cdots\oplus_{2}[-u_{m},u_{m}]|^{2}=|UB_{2}^{m}|^{2}_{n}={\mathop{\rm
det}}(UU^{*})|B_{2}^{n}|^{2}=|B_{2}^{n}|^{2}\sum_{|I|=n}({\mathop{\rm
det}}(u_{i})_{i\in I})^{2}.$ (47)

Next, using

$P_{u^{\bot}}([-u_{1},u_{1}]\oplus_{2}\cdots\oplus_{2}[-u_{m},u_{m}])=[-P_{u^{\bot}}u_{1},P_{u^{\bot}}u_{1}]\oplus_{2}\cdots\oplus_{2}[-P_{u^{\bot}}u_{m},P_{u^{\bot}}u_{m}],$

we get that

$|P_{u^{\bot}}([-u_{1},u_{1}]\oplus_{2}\cdots\oplus_{2}[-u_{m},u_{m}])|^{2}=|B_{2}^{n-1}|^{2}\sum_{|I|=n-1}({\mathop{\rm
det}}(P_{u^{\bot}}u_{i})_{i\in I})^{2}.$

Thus our goal is to prove that, for $N\geq N^{\prime}$ and
$A=[-u_{1},u_{1}]\oplus_{2}\cdots\oplus_{2}[-u_{N^{\prime}},u_{N^{\prime}}]$
and
$B=[-u_{N^{\prime}+1},u_{N^{\prime}+1}]\oplus_{2}\cdots\oplus_{2}[-u_{N},u_{N}]$
we have

$\displaystyle\frac{\sum\limits_{M\subset\\{1,\dots,N^{\prime}\\},|M|=n}|{\mathop{\rm
det}}(\\{u_{m}\\}_{m\in
M}|^{2}}{\sum\limits_{M\subset\\{1,\dots,N^{\prime}\\},|M|=n-1}|{\mathop{\rm
det}}(\\{P_{e_{1}^{\perp}}u_{m}\\}_{m\in M})|^{2}}$
$\displaystyle+\frac{\sum\limits_{L\subset\\{N^{\prime}+1,\dots,N\\},|L|=n}|{\mathop{\rm
det}}(\\{u_{l}\\}_{l\in
L}|^{2}}{\sum\limits_{L\subset\\{N^{\prime}+1,\dots,N\\},|L|=n-1}|{\mathop{\rm
det}}(\\{P_{e_{1}^{\perp}}u_{l}\\}_{l\in L}|^{2}}$
$\displaystyle\leq\frac{\sum\limits_{M\subset\\{1,\dots,N\\},|M|=n}|{\mathop{\rm
det}}(\\{u_{m}\\}_{m\in
M}|^{2}}{\sum\limits_{M\subset\\{1,\dots,N\\},|M|=n-1}|{\mathop{\rm
det}}(\\{P_{e_{1}^{\perp}}(u_{m})\\}_{m\in M}|^{2}}.$ (48)

For $I\subset\\{1,\dots,m\\}$ let $U_{I}$ be the $n\times n$ submatrix built
from $U$ by taking the columns with indices in $I$. Denote by
$z_{1},\dots,z_{n}\in{\mathbb{R}}^{m}$ the rows of the matrix $U$. Since
$n\leq m$, the set $\sum_{i=1}^{n}[0,z_{i}]$ is a parallelotope leaving in a
$n$-dimensional subspace of ${\mathbb{R}}^{m}$. Thus,

$\left|\sum_{i=1}^{n}[0,z_{i}]\right|_{n}^{2}=\sum_{|I|=n}({\mathop{\rm det}}\
U_{I})^{2}.$

Therefore, we get that, for $m\geq n$, an $n\times m$ matrix $U$ whose rows
are $z_{1},\dots,z_{n}\in{\mathbb{R}}^{m}$ and columns
$u_{1},\dots,u_{m}\in{\mathbb{R}}^{n}$:

$|[-u_{1},u_{1}]\oplus_{2}\cdots\oplus_{2}[-u_{m},u_{m}]|_{n}=|B_{2}^{n}|\left|\sum_{i=1}^{n}[0,z_{i}]\right|_{n}.$

Let us write $U$ as a block matrix with two blocks $U=(U_{1}|0)+(0|U_{2})$,
with $U_{1}$ being an $n\times k$ matrix and $U_{2}$ an $n\times(m-k)$ matrix
and we denote $V=(U_{1}|0)$ and $W=(0|U_{2})$. Moreover, denote by
$U^{\prime},V^{\prime},W^{\prime}$ the matrices obtained from $U,V,W$ by
erasing the $n^{th}$ row. Then we only need to prove that

$\frac{\sum_{|I|=n}({\mathop{\rm det}}\
U_{I})^{2}}{\sum_{|I|=n-1}({\mathop{\rm det}}\
U^{\prime}_{I})^{2}}\geq\frac{\sum_{|I|=n}({\mathop{\rm det}}\
V_{I})^{2}}{\sum_{|I|=n-1}({\mathop{\rm det}}\
V^{\prime}_{I})^{2}}+\frac{\sum_{|I|=n}({\mathop{\rm det}}\
W_{I})^{2}}{\sum_{|I|=n-1}({\mathop{\rm det}}\ W^{\prime}_{I})^{2}}.$

Recall that $z_{1},\dots,z_{n}\in{\mathbb{R}}^{m}$ are the rows of $U$. Thus
the rows of $V$ are $P_{E}z_{1},\dots,P_{E}z_{n}$, where $P_{E}$ denotes the
projection on the first $k$ coordinates and the rows of $W$ are
$P_{E^{\bot}}z_{1},\dots,P_{E^{\bot}}z_{n}$, where $P_{E^{\bot}}$ denotes the
projection on the $n-k$ last coordinates. Thus we only need to show the
following relationship for low-dimensional parallelotopes

$\left(\frac{\left|\sum_{i=1}^{n}[0,z_{i}]\right|}{\left|\sum_{i=1}^{n-1}[0,z_{i}]\right|}\right)^{2}\geq\left(\frac{\left|\sum_{i=1}^{n}[0,P_{E}z_{i}]\right|}{\left|\sum_{i=1}^{n-1}[0,P_{E}z_{i}]\right|}\right)^{2}+\left(\frac{\left|\sum_{i=1}^{n}[0,P_{E^{\bot}}z_{i}]\right|}{\left|\sum_{i=1}^{n-1}[0,P_{E^{\bot}}z_{i}]\right|}\right)^{2}.$

Using that the volume of a parallelotope is the product of the volume of one
of its face and its height, we get that, if
$H_{n}={\mathrm{span}}(z_{1},\dots,z_{n-1})$, then

$\frac{\left|\sum_{i=1}^{n}[0,z_{i}]\right|}{\left|\sum_{i=1}^{n-1}[0,z_{i}]\right|}=d(z_{n},H_{n}).$

So we are reduced to prove that

$d(z_{n},H_{n})^{2}\geq
d(P_{E}z_{n},P_{E}H_{n})^{2}+d(P_{E^{\bot}}z_{n},P_{E^{\bot}}H_{n})^{2}.$

Let $h_{n}\in H_{n}$ such that $d(z_{n},H_{n})=|z_{n}-h_{n}|$. By Pythagoras’
theorem,

$d(z_{n},H_{n})^{2}=|z_{n}-h_{n}|^{2}=\|P_{E}z_{n}-P_{E}h_{n}\|^{2}+\|P_{E^{\bot}}z_{n}-P_{E^{\bot}}h_{n}\|^{2}.$

Since $P_{E}h_{n}\in P_{E}H_{n}$ and $P_{E^{\bot}}z_{n}\in P_{E^{\bot}}H_{n}$,
we conclude. $\Box$

Second proof of the inequality: We give another proof of Theorem 6.2, using
the comparison of the $\ell_{2}$ sum and the radial $2$-sum. For any symmetric
convex bodies $K$ and $L$ in ${\mathbb{R}}^{n}$, one has

$K\oplus_{2}L\supset K\widetilde{+}_{2}L:=\cup_{u\in
S^{n-1}}\left[0,\sqrt{\rho_{K}^{2}(u)+\rho_{L}^{2}(u)}u\right].$ (49)

Indeed, using support functions, the fact that $K\supset\rho_{K}(u)[-u,u]$
implies that $h_{K}(x)\geq\rho_{K}(u)|\langle x,u\rangle|$, for any
$x\in{\mathbb{R}}^{n}$ and thus

$h_{K\oplus_{2}L}(x)=\sqrt{h_{K}(x)^{2}+h_{L}(x)^{2}}\geq\sqrt{\rho_{K}(u)^{2}+\rho_{L}(u)^{2}}|\langle
x,u\rangle|=\sqrt{\rho_{K}(u)^{2}+\rho_{L}(u)^{2}}h_{[-u,u]}(x).$

The formula (49) can be also restated in the language of radial functions:

$\rho_{K\oplus_{2}L}(u)\geq\sqrt{\rho_{K}^{2}(u)+\rho_{L}^{2}(u)},\mbox{ for
all }u\in S^{n-1}.$ (50)

Next, we notice nice a formula for the volume of the orthogonal hyperplane
projection of an ellipsoid (see, for example, [13, 41]), to which we give a
very simple proof. Let $\mathcal{E}=TB_{2}^{n},$ for some positive definite
$T$ then

$\frac{|P_{u^{\bot}}\mathcal{E}|}{|\mathcal{E}|\,\|u\|_{\mathcal{E}}}=\frac{nV(TB_{2}^{n}[n-1],[0,u])}{|TB_{2}^{n}|\,|T^{-1}u|}=\frac{nV(B_{2}^{n}[n-1],[0,T^{-1}u])}{|B_{2}^{n}|\,|T^{-1}u|}=\frac{|B_{2}^{n-1}|}{|B_{2}^{n}|}.$
(51)

Using the above we get

$\frac{|\mathcal{E}|}{|P_{u}\mathcal{E}|}=\frac{|B_{2}^{n}|}{|B_{2}^{n-1}|}\rho_{\mathcal{E}}(u).$
(52)

Thus, using this formula and (49), we deduce that for any ellipsoids $A,B$

$\displaystyle\frac{|A\oplus_{2}B|^{2}}{|P_{u}(A\oplus_{2}B)|^{2}}$
$\displaystyle=\frac{|B_{2}^{n}|^{2}}{|B_{2}^{n-1}|^{2}}\rho_{A\oplus_{2}B}(u)^{2}$
$\displaystyle\geq\frac{|B_{2}^{n}|^{2}}{|B_{2}^{n-1}|^{2}}(\rho_{A}(u)^{2}+\rho_{B}(u)^{2})=\frac{|A|^{2}}{|P_{u}A|^{2}}+\frac{|B|^{2}}{|P_{u}B|^{2}}.$
(53)

$\Box$

Proof of the equality case: This second proof also helps us to treat the
equality case. From (52) there is equality if and only if

$\rho_{A\oplus_{2}B}(u)^{2}=\rho_{A}(u)^{2}+\rho_{B}(u)^{2}.$

The above is equivalent to the
$(\rho_{A}(u)^{2}+\rho_{B}(u)^{2})^{1/2}u\in\partial(A\oplus_{2}B)$. From here
we get that, if $n$ is a normal vector to $\partial(A\oplus_{2}B)$ at
$(\rho_{A}(u)^{2}+\rho_{B}(u)^{2})^{1/2}u$ then

$h_{A\oplus_{2}B}(n)=(\rho_{A}(u)^{2}+\rho_{B}(u)^{2})^{1/2}\langle
u,n\rangle.$

The above is equivalent to $h_{A}(n)=\rho_{A}(u)\langle u,n\rangle\mbox{ and
}h_{B}(n)=\rho_{B}(u)\langle u,n\rangle,$ or simply to say that the normal
vector to $\partial A$ at $\rho_{A}(u)u$ is the normal vector to $\partial B$
at $\rho_{B}(u)u$. $\Box$

######  Remark 6.3.

If in the proof of equality case in the Theorem 6.2 we would represent
$A=T_{1}B_{2}^{n}$ and $B=T_{2}B_{2}^{n},$ where $T_{1},T_{2}$ are two
positive symmetric matrices. Then $\|x\|^{2}_{A}=\langle
T_{1}^{-2}x,x\rangle.$ and the normal vector to $\partial A$ at $\rho_{A}(u)u$
is parallel to $T_{1}^{-2}u$, and the similar statement is true for $B$. Thus
our condition on parallel tangent hyperplanes is equivalent to the fact that
there is $\lambda>0$ such that $(T_{1}^{-2}-\lambda T_{2}^{-2})u=0$ or simply
that $u$ is an eigenvector for matrix $T_{1}^{2}T_{2}^{-2}$.

######  Remark 6.4.

Observe that in Theorem 6.2 there is an equality at least for $n$ directions
$u$ (not counting $-u$) and there is equality for every $u$ if and only if two
ellipsoids are homothetic.

Let us now present some consequences or Theorem 6.2.

######  Corollary 6.5.

Let $n$ be a positive integer. Let $A$ and $B$ be $L_{2}$-zonoids in
${\mathbb{R}}^{n}$ and $u$ in $S^{n-1}$. Then, the function $h$ defined, for
$t\geq 0$, by

$h_{2}(t)=\frac{|A\oplus_{2}(\sqrt{t}B)|^{2}}{|P_{u^{\bot}}(A\oplus_{2}(\sqrt{t}B))|^{2}_{n-1}}$

is concave on ${\mathbb{R}}_{+}$.

###### Proof.

For any $\lambda\in[0,1]$, one has
$A=(\sqrt{1-\lambda}A)\oplus_{2}(\sqrt{\lambda}A)$. Thus one deduces that

$A\oplus_{2}\left(\sqrt{(1-\lambda)s+\lambda
t}B\right)=\left(\sqrt{1-\lambda}(A\oplus_{2}\sqrt{s}B)\right)\oplus_{2}\left(\sqrt{\lambda}(A\oplus_{2}\sqrt{t}B)\right).$

Using Theorem 6.2 and the homogeneity of volume, we deduce that $h_{2}$ is
concave.  $\Box$

Next we show that Theorem 6.2 has the following additional applications:

######  Theorem 6.6.

Let $n$ be an integer, then for any $1\leq k\leq n$ and for every pair of
$L_{2}$-zonoids $A$ and $B$ in ${\mathbb{R}}^{n}$ and any ($n-k$)-dimensional
subspace $E$ of ${\mathbb{R}}^{n}$ one has

$\left(\frac{|A\oplus_{2}B|}{|P_{E}(A\oplus_{2}B)|_{n-k}}\right)^{\frac{2}{k}}\geq\left(\frac{|A|}{|P_{E}A|_{n-k}}\right)^{\frac{2}{k}}+\left(\frac{|B|}{|P_{E}B|_{n-k}}\right)^{\frac{2}{k}}.$
(54)

###### Proof.

The proof goes by induction on $k$. Theorem 6.2 establishes the case $k=1$ and
any $n\geq 1$. We assume that the inequality holds for some $1\leq k\leq n-1$
for all $L_{2}$-zonoids $A,B$ in ${\mathbb{R}}^{n}$ and all $n-k$ dimensional
subspace of ${\mathbb{R}}^{n}$. Let $E$ be a $n-k-1$ dimensional subspace of
${\mathbb{R}}^{n}$. Then one may write $E=F\cap u^{\bot}$, for some $n-k$
dimensional subspace $F$ and $u\in F^{\bot}$. Then, applying Theorem 6.2 to
$P_{F}A$ and $P_{F}B$ and using that $P_{u^{\bot}}\circ P_{F}=P_{E}$ we get

$\left(\frac{|P_{F}(A\oplus_{2}B)|_{n-k}}{|P_{E}(A\oplus_{2}B)|_{n-k-1}}\right)^{2}\geq\left(\frac{|P_{F}A|_{n-k}}{|P_{E}A|_{n-k-1}}\right)^{2}+\left(\frac{|P_{F}B|_{n-k}}{|P_{E}B|_{n-k-1}}\right)^{2}.$
(55)

Applying (54) to $E=F$, raising the equation to power $k/2$ and taking the
product with (55), we get

$\displaystyle\frac{|A\oplus_{2}B|}{|P_{E}(A\oplus_{2}B)|_{n-k-1}}\geq$ (56)
$\displaystyle\left(\left(\frac{|P_{F}A|_{n-k}}{|P_{E}A|_{n-k-1}}\right)^{2}+\left(\frac{|P_{F}B|_{n-k}}{|P_{E}B|_{n-k-1}}\right)^{2}\right)^{\frac{1}{2}}\left(\left(\frac{|A|}{|P_{F}A|_{n-k}}\right)^{\frac{2}{k}}+\left(\frac{|B|}{|P_{F}B|_{n-k}}\right)^{\frac{2}{k}}\right)^{\frac{k}{2}}.$

From Hölder’s inequality, we conclude that

$\frac{|A\oplus_{2}B|}{|P_{E}(A\oplus_{2}B)|_{n-k-1}}\geq\left(\left(\frac{|A|}{|P_{E}A|_{n-k-1}}\right)^{\frac{2}{k+1}}+\left(\frac{|B|}{|P_{E}B|_{n-k-1}}\right)^{\frac{2}{k+1}}\right)^{\frac{k+1}{2}},$

which is (54) for $k+1$.  $\Box$

######  Corollary 6.7.

Let $k,n$ be a integer with $1\leq k\leq n.$ Then for every $L_{2}$-zonoids
$A$ and $B$ in ${\mathbb{R}}^{n}$ and every zonoids $Z_{1},\dots,Z_{k}$ in
${\mathbb{R}}^{n},$

$\displaystyle\left(\frac{|A\oplus_{2}B|}{V((A\oplus_{2}B)[n-k],Z_{1},\dots,Z_{k})}\right)^{\frac{2}{k}}\geq$
(57)
$\displaystyle\left(\frac{|A|}{V(A[n-k],Z_{1},\dots,Z_{k})}\right)^{\frac{2}{k}}+\left(\frac{|B|}{V(B[n-1],Z_{1},\dots,Z_{k})}\right)^{\frac{2}{k}},$

and thus the function $f$ defined, for $t\geq 0$, by

$f(t)=\frac{|A\oplus_{2}\sqrt{t}B|^{\frac{2}{k}}}{|V((A\oplus_{2}\sqrt{t}B)[n-k],Z_{1},\dots,Z_{k})|^{\frac{2}{k}}_{n-k}}$

is concave on ${\mathbb{R}}_{+}$. Moreover

$\frac{|A\oplus_{2}B|^{2}}{|\partial(A\oplus_{2}B)|^{2}}\geq\frac{|A|^{2}}{|\partial
A|^{2}}+\frac{|B|^{2}}{|\partial B|^{2}},$ (58)

and thus the function $g$ defined, for $t\geq 0$, by

$g(t)=\frac{|A\oplus_{2}\sqrt{t}B|^{2}}{|\partial(A\oplus_{2}\sqrt{t}B)|^{2}_{n-1}}$

is concave on ${\mathbb{R}}_{+}$.

###### Proof.

First notice that Theorem 6.6 may be reformulated in the following way. Let
$u_{1},\dots,u_{k}$ be an orthonormal system in ${\mathbb{R}}^{n}$. Then using
(15) we have

$\displaystyle\left(\frac{|A\oplus_{2}B|}{V((A\oplus_{2}B)[n-k],[0,u_{1}],\dots,[0,u_{k}])}\right)^{\frac{2}{k}}\geq$
(59)
$\displaystyle\left(\frac{|A|}{V(A[n-k],[0,u_{1}],\dots,[0,u_{k}])}\right)^{\frac{2}{k}}$
$\displaystyle+\left(\frac{|B|}{V(B[n-k],[0,u_{1}],\dots,[0,u_{k}])}\right)^{\frac{2}{k}}.$

Applying a linear transform, (59) holds for any linearly independent system
$u_{1},\dots,u_{k}$. Then, for $x,y\geq 0$, define

$\varphi(x,y)=(x^{-\frac{2}{k}}+y^{-\frac{2}{k}})^{-\frac{k}{2}}=\|(x,y)\|_{-\frac{2}{k}}.$
(60)

For a compact convex set $A$ and $u_{1},\dots,u_{k}\in{\mathbb{R}}^{n}$, let

$\psi_{A}(u_{1},\dots,u_{k})=\frac{V(A[n-k],[0,u_{1}],\dots,,[0,u_{k}])}{|A|}.$
(61)

From (59) we know that, if $u_{1},\dots,u_{k}$ are linerly independent, then

$\psi_{A\oplus_{2}B}(u_{1},\dots,u_{k})\leq\varphi(\psi_{A}(u_{1},\dots,u_{k}),\psi_{B}(u_{1},\dots,u_{k})).$

For $i=1,\dots,k$, let
$Z_{i}=[0,u_{i,1}]\oplus_{2}\cdots\oplus_{2}[0,u_{i,m_{i}}]$ be a
$2$-zonotope. Assume that for any set of distinct $k$ vectors from the set
$\\{u_{i,j}\\}$ is an independent sequence (this can be achieved by a small
perturbation of vectors $u_{i,j}$). Using that $\varphi$, being a
$-\frac{2}{k}$-norm, satisfies the reverse Minkowski inequality, we deduce
that

$\displaystyle\sum_{i=1}^{k}\sum_{j_{i}=1}^{m_{i}}\psi_{A\oplus_{2}B}(u_{1,j_{1}},\dots,u_{k,j_{k}})$
$\displaystyle\leq\sum_{i=1}^{k}\sum_{j_{i}=1}^{m_{i}}\varphi(\psi_{A}(u_{1,j_{1}},\dots,u_{k,j_{k}}),\psi_{B}(u_{1,j_{1}},\dots,u_{k,j_{k}}))$
$\displaystyle\leq\varphi\left(\sum_{i=1}^{k}\sum_{j_{i}=1}^{m_{i}}\psi_{A}(u_{1,j_{1}},\dots,u_{k,j_{k}}),\sum_{i=1}^{k}\sum_{j_{i}=1}^{m_{i}}\psi_{B}(u_{1,j_{1}},\dots,u_{k,j_{k}})\right).$

Thus we get

$\frac{V((A\oplus_{2}B)[n-k],Z_{1},\dots,Z_{k})}{|A\oplus_{2}B|}\leq\varphi\left(\frac{V(A[n-k],Z_{1},\dots,Z_{k})}{|A|},\frac{V(B[n-k],Z_{1},\dots,Z_{k})}{|B|}\right),$

which is (57), for $Z_{1},\dots,Z_{k}$ being a zonotopes. The result for
zonoids follows by approximation. We apply (57) with $k=1$ and
$Z_{1}=B_{2}^{n}$ to prove (58). The concavity of $f$ and $g$ is proved with
the method used for Corollary 6.5.

$\Box$

###  6.2 The case $p\neq 2$

######  Proposition 6.8.

The answer to Question 6.1 is negative when $p>2$ and $n\geq 2$: a
counterexample is given by $A=B_{2}^{n}$ and
$B=\varepsilon^{\frac{1}{p}}[-v,v]$, for some $\varepsilon>0$.

###### Proof.

Let $p>2$. We disprove the weaker statement

$\frac{|A\oplus_{p}B|}{|P_{u^{\bot}}(A\oplus_{p}B)|_{n-1}}\geq\frac{|A|}{|P_{u^{\bot}}A|_{n-1}}.$
(62)

We restate (62) with $B=\varepsilon^{\frac{1}{p}}[-v,v],$ $v\in S^{n-1}$ and
$\varepsilon\to 0.$ For this we use the extension of the classical notion of
the mixed volumes introduced by Lutwak [31, 32], who proved that

$\lim\limits_{\varepsilon\to
0}\frac{|A\oplus_{p}(\varepsilon^{\frac{1}{p}}\,B)|-|A|}{\varepsilon}=\frac{1}{p}\int\limits_{S^{n-1}}h^{p}_{B}(u)h_{A}^{1-p}dS_{A}(u),$
(63)

for $p>1$ and all convex compact sets $A,B,$ containing the origin and defined

$V_{p}(A[n-1],B)=\frac{1}{n}\int\limits_{S^{n-1}}h^{p}_{B}(u)h_{A}^{1-p}dS_{A}(u).$
(64)

Taking $B=\varepsilon^{\frac{1}{p}}[-v,v]$ for some $v\in S^{n-1},$ we get

$\Big{|}A\oplus_{p}B\Big{|}=|A|+\frac{\varepsilon}{p}\int\limits_{S^{n-1}}|\langle
v,x\rangle|^{p}h_{A}^{1-p}dS_{A}(x)+o(\varepsilon)$

Assuming $v$ in $u^{\perp},$ we get

$\Big{|}P_{u^{\perp}}A\oplus_{p}P_{u^{\perp}}B\Big{|}=|P_{u^{\perp}}A|+\frac{\varepsilon}{p}\int\limits_{S^{n-1}\cap
u^{\perp}}|\langle
v,x\rangle|^{p}h_{A}^{1-p}(x)dS_{P_{u^{\perp}}A}(x)+o(\varepsilon).$

Assume, by way of contradiction, that (62) is true for all $L_{p}$-zonoids
$A\subset{\mathbb{R}}^{n}.$ Then,

$|A|\int\limits_{S^{n-1}\cap u^{\perp}}|\langle
v,x\rangle|^{p}h_{A}^{1-p}(x)dS_{P_{u^{\perp}}A}(x)\leq|P_{u^{\perp}}A|\int\limits_{S^{n-1}}|\langle
v,x\rangle|^{p}h_{A}^{1-p}(x)dS_{A}(x).$ (65)

Now we take $A=B_{2}^{n}$, $u=e_{2}$ and $v=e_{1}$. Notice that $B_{2}^{n}$ is
an $L_{p}$ zonoid for all $p\geq 1$. Next (65) becomes

$|B_{2}^{n}|\int_{S^{n-2}}|x_{1}|^{p}dS_{B_{2}^{n-1}}(x)\leq|B_{2}^{n-1}|\int_{S^{n-1}}|x_{1}|^{p}dS_{B_{2}^{n}}(x).$
(66)

Using polar coordinates and Fubini’s theorem, we get

$\int_{S^{n-1}}|x_{1}|^{p}dS_{B_{2}^{n}}(x)=(n+p)\int_{B_{2}^{n-1}}|z_{1}|^{p}dz=\frac{2\pi^{\frac{n-1}{2}}\Gamma\left(\frac{p+1}{2}\right)}{\Gamma\left(\frac{p+n}{2}\right)}.$

Thus (66) becomes

$\Gamma\left(\frac{n+1}{2}\right)\Gamma\left(\frac{p+n}{2}\right)\leq\Gamma\left(\frac{n+2}{2}\right)\Gamma\left(\frac{p+n-1}{2}\right).$

Using the strict $\log$-convexity of $\Gamma$ function the above is only true
if and only if $p\leq 2$.  $\Box$

######  Proposition 6.9.

Fix $p>1$. Then (46) does not hold in the class of all convex symmetric bodies
in ${\mathbb{R}}^{n}$.

###### Proof.

Let us first construct an example in ${\mathbb{R}}^{2}$. We note that

$\Big{|}P_{u^{\perp}}A\oplus_{p}P_{u^{\perp}}\varepsilon^{\frac{1}{p}}[-v,v]\Big{|}=2h_{A}(v)+\frac{2\varepsilon}{p}h_{A}(v)^{1-p}+o(\varepsilon).$

Thus (62), for $n=2$, would imply

$|A|\leq h_{A}(v)^{p}\int\limits_{S^{1}}|\langle
v,x\rangle|^{p}h_{A}^{1-p}(x)dS_{A}(x).$ (67)

Consider $a\in(0,1),$ let $A=\\{(x_{1},x_{2})\in{\mathbb{R}}^{2},|x_{i}|\leq
1,|x_{1}\pm x_{2}|\leq 2-a\\}.$ Then $|A|=4-2a^{2}$. We check (67) with
$v=e_{1}$. We first note that $h_{A}(e_{1})=1$. Next we compute

$f(x)=|\langle e_{1},x\rangle|^{p}h_{A}^{1-p}(x)S_{A}(x)$

for different normal vectors $x$ of $A$. We first note that $f(\pm e_{2})=0$,
$f(\pm e_{1})=2(1-a)$ and $f((\pm 1/\sqrt{2},\pm 1/\sqrt{2}))=a(2-a)^{1-p}.$
Thus to contradict (67) we must select $a$ such that

$4-2a^{2}>4(1-a)+4a(2-a)^{1-p},$

or $a<2-2^{1/p},$ which is possible for every $p>1$.

To build a counterexample in ${\mathbb{R}}^{n}$ for $n\geq 3$, we use the fact
that if $K$ is a convex body in ${\rm span}\\{e_{1},\dots,e_{n_{1}}\\}$ and
$L$ is a convex body in ${\rm span}\\{e_{n_{1}+1},\dots,e_{n_{1}+n_{2}}\\}$,
then

$|K\oplus_{p}L|=\frac{\Gamma(\frac{n_{1}}{q}+1)\Gamma(\frac{n_{2}}{q}+1)}{\Gamma(\frac{n_{1}+n_{1}}{q}+1)}|K||L|=c_{n_{1},n_{2},q}|K||L|,$
(68)

where $1/p+1/q=1$. Let $A_{n}=B_{q}^{n-2}\oplus_{p}A_{2}$, where
$A_{2}\subset{\rm span}\\{e_{1},e_{2}\\}$ is the counterexample created above,
and $B^{n-2}_{q}\subset{\rm span}\\{e_{3},\dots,e_{n}\\}$. Then (68) gives

$\frac{|A_{n}\oplus_{p}[-v,v]|}{|P_{e_{1}^{\bot}}(A_{n}\oplus_{p}[-v,v])|_{n-1}}=\frac{|B_{q}^{n-2}\oplus_{p}(A_{2}\oplus_{p}[-v,v])|}{|B_{q}^{n-2}\oplus_{p}P_{e_{1}^{\bot}}(A_{2}\oplus_{p}[-v,v])|_{n-1}}<\frac{|A_{n}|}{|P_{e_{1}^{\bot}}A_{n}|_{n-1}}.$

$\Box$

Let us note that the direct interpretation of the volume of $L_{p}$-zonotopes
in terms of determinants is only possible in the case when $p=1$ or $p=2$
[28], thus it is natural to ask if the determinant inequality that we proved
in the case $p=2$ is still true in the case $p\not=2:$

######  Question 6.10.

Let $p\geq 1$, consider $N\geq N^{\prime}\geq n$ and a sequence for vectors
$\\{u_{i}\\}_{i=1}^{N}$ in ${\mathbb{R}}^{n}$ is it true that

$\displaystyle\frac{\sum\limits_{M\subset\\{1,\dots,N^{\prime}\\},|M|=n}|{\mathop{\rm
det}}(\\{u_{m}\\}_{m\in
M}|^{p}}{\sum\limits_{M\subset\\{1,\dots,N^{\prime}\\},|M|=n-1}|{\mathop{\rm
det}}(\\{P_{e_{1}^{\perp}}u_{m}\\}_{m\in M})|^{p}}$
$\displaystyle+\frac{\sum\limits_{L\subset\\{N^{\prime}+1,\dots,N\\},|L|=n}|{\mathop{\rm
det}}(\\{u_{l}\\}_{l\in
L}|^{p}}{\sum\limits_{L\subset\\{N^{\prime}+1,\dots,N\\},|L|=n-1}|{\mathop{\rm
det}}(\\{P_{e_{1}^{\perp}}u_{l}\\}_{l\in L}|^{p}}$
$\displaystyle\leq\frac{\sum\limits_{M\subset\\{1,\dots,N\\},|M|=n}|{\mathop{\rm
det}}(\\{u_{m}\\}_{m\in
M}|^{p}}{\sum\limits_{M\subset\\{1,\dots,N\\},|M|=n-1}|{\mathop{\rm
det}}(\\{P_{e_{1}^{\perp}}(u_{m})\\}_{m\in M}|^{p}}?$ (69)

Notice that the above question also has a negative answer when $p>2$. Let
$N=n=2,$ $N^{\prime}=1$ and the matrix

$\begin{pmatrix}1&-1&0\\\ 1&1&1\end{pmatrix}.$

Then (69) becomes

$\frac{2^{p}}{1+1}\leq\frac{2^{p}+1+1}{1+1+1},$

which is false for $p>2$.

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Matthieu Fradelizi
Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, LAMA UMR8050 F-77447 Marne-
la-Vallée, France
E-mail address<EMAIL_ADDRESS>
Mokshay Madiman
University of Delaware
Department of Mathematical Sciences
501 Ewing Hall
Newark, DE 19716, USA
E-mail address<EMAIL_ADDRESS>
Mathieu Meyer
Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, LAMA UMR8050 F-77447 Marne-
la-Vallée, France
E-mail address<EMAIL_ADDRESS>
Artem Zvavitch
Department of Mathematical Sciences
Kent State University
Kent, OH 44242, USA
E-mail address<EMAIL_ADDRESS>