# Isoperimetric problems for zonotopes

Antal Joós  and  Zsolt Lángi  Antal Joós
Department of Mathematics, University of Dunaújváros, Táncsics M. u. 1/a,
Dunaújváros, Hungary, 2400<EMAIL_ADDRESS>Zsolt Lángi
Department of Geometry, Budapest University of Technology, Egry József utca
1., Budapest, Hungary, 1111 and
MTA-BME Morphodynamics Research Group, Műegyetem rkp. 3., Budapest, Hungary,
1111<EMAIL_ADDRESS>
###### Abstract.

Shephard (Canad. J. Math. 26: 302-321, 1974) proved a decomposition theorem
for zonotopes yielding a simple formula for their volume. In this note we
prove a generalization of this theorem yielding similar formulas for their
intrinsic volumes. We use this result to investigate geometric extremum
problems for zonotopes generated by a given number of segments. In particular,
we solve isoperimetric problems for $d$-dimensional zonotopes generated by $d$
or $d+1$ segments, and give asymptotic estimates for the solutions of similar
problems for zonotopes generated by sufficiently many segments. In addition,
we present applications of our results to the $\ell_{1}$ polarization problem
on the unit sphere and to a vector-valued Maclaurin inequality conjectured by
Brazitikos and McIntyre in 2021.

###### Key words and phrases:

zonotope, parallelotope, rhombic dodecahedron, isoperimetric problem,
intrinsic volume, polarization problem

###### 2010 Mathematics Subject Classification:

52B60, 52A40, 52A39

Partially supported by the BME Water Sciences & Disaster Prevention TKP2020
Institution Excellence Subprogram, grant no. TKP2020 BME-IKA-VIZ and the NKFIH
grant K134199.

##  1\. Introduction

An important family of convex polytopes in the $d$-dimensional Euclidean space
$\mathbb{R}^{d}$ is the class of so-called _zonotopes_ , polytopes obtained as
Minkowski sums of finitely many segments. Zonotopes have been in the focus of
research since the middle of the 20th century, earning a separate chapter in
the famous book [16] of Coxeter in 1943 and a place in the problem collection
[44] of Shephard. They are connected to various branches of mathematics, as an
example, we may mention the fact that their face lattices correspond to the
combinatorial classes of central hyperplane arrangements in $\mathbb{R}^{d+1}$
[25, 47]. They play a central role in the theory of projections of convex
polytopes, both being the projections of affine cubes (see e.g. [37]), and, by
Cauchy’s projection formula [21], being the projection bodies of convex
polytopes in $\mathbb{R}^{d}$ (see also [31]). They are closely related to
parallelohedra, the convex polytopes whose translates fill the space (see [18,
36]), and appear in lattice geometry (see e.g. [32, 28, 4] or the survey
[14]). Zonotopes are also investigated and often applied outside pure
mathematics [1, 5, 6, 26].

We note that there is a rich literature about isoperimetric type problems for
zonotopes. From amongst these papers, we mention the paper [9] of Bezdek
comparing the inradius of a rhombic dodecahedron to its intrinsic volumes,
[19] of Filliman comparing the volume of a zonotope to the total squared
lengths of its generating vectors, and [17] of Deza, Pournin and Sukegawa
giving a sharp asymptotic estimate on the maximum diameter of a primitive
zonotope. Recently, Fradelizi et al. [20] investigated various volume
inequalities for the Minkowski sums of zonoids. For open problems regarding
zonotopes, the interested reader is referred also to [30].

An elegant, simple formula exists for the volume of a zonotope in terms of its
generating segments; this formula is usually attributed to McMullen [35] or
Shephard [45] and follows e.g. from a decomposition theorem for zonotopes in
[45]. In [22], this formula was extended to zonotopes whose dimension is
strictly less than that of the ambient space. Using an integral geometric
approach, in a very recent paper Brazitikos and McIntyre [15] found a
generalization of this formula for all intrinsic volumes of a zonotope.

As a preliminary result, we prove a generalization of Shephard’s decomposition
theorem reproving the formulas for the intrinsic volumes of a zonotope in
[15]. Our proof is presented in Section 2, where we also collect all the tools
required in our investigation. In the main part of the paper we prove
isoperimetric inequalities for zonotopes. We focus on two types of zonotopes.
In particular, we investigate zonotopes in $\mathbb{R}^{d}$ generated by $d$
or $d+1$ segments, and also zonotopes in $\mathbb{R}^{d}$ generated by $n\gg
d$ segments. We collect our results for these two types of zonotopes in
Sections 3 and 4, respectively.

In the paper, we also point out applications of our results to other problems
in mathematics. We focus on two problems, one of which is the so-called
_$\ell_{p}$ -polarization problem_ discussed e.g. in [3] (see also [39]), and
the other one is a generalization of the well known Maclaurin inequality [10]
by Brazitikos and McIntyre [15], comparing the values of elementary symmetric
functions evaluated at positive real numbers. We introduce these problems and
their relations to zonotopes in detail in Section 2, and describe our related
results at their appropriate places in Sections 3 and 4.

Finally, in Section 5 we collect some remarks and open questions.

##  2\. Preliminaries

In our investigation, we denote the origin of the space $\mathbb{R}^{n}$ by
$o$. We denote the closed unit ball centered at $o$ by $\mathbf{B}^{d}$, and
its boundary and volume by $\mathbb{S}^{d-1}$ and $\kappa_{d}$, respectively,
where we set $\kappa_{0}=1$. For any points $p,q\in\mathbb{R}^{d}$, we denote
the closed segment with endpoints $p,q$ by $[p,q]$, and the Euclidean norm of
$p$ by $|p|$. We call a compact, convex set with non-empty interior a _convex
body_.

Let $K\subset\mathbb{R}^{d}$ be a compact, convex set. By Steiner’s formula,
the volume of the convex body $K+t\mathbf{B}^{d}$ is a polynomial of $t$ of
degree $d$. We write this polynomial in the form

(1) $V_{d}(K+t\mathbf{B}^{d})=\sum_{i=0}^{d}\kappa_{d-i}V_{i}(K)t^{d-i}.$

For $i=1,2,\ldots,n$, the quantity $V_{i}(K)$ is called the _$i$ th intrinsic
volume of $K$_ [21]. We recall that if the dimension of $K$ is $m$ with $1\leq
m\leq d$, then for any $i>m$, $V_{i}(K)=0$, and $V_{m}(K)$ is equal to the
$m$-dimensional volume (Lebesgue measure) of $K$. Finally, if
$K\subset\mathbb{R}^{d}$ is a convex body, we denote the surface area and the
mean width of $K$ by $\operatorname{surf}(K)$ and w(K), respectively.

It is a well known fact that for any convex body $K\subset\mathbb{R}^{d}$
there is a unique ball of minimal radius containing $K$. We call this ball the
_circumball_ of $K$, its radius the _circumradius_ of $K$, which we denote by
$\operatorname{cr}(K)$. Similarly, we call a largest ball contained in $K$ an
_inball_ of $K$, and denote its radius by $\operatorname{ir}(K)$. Finally, for
any $v_{1},v_{2},\ldots,v_{d}\in\mathbb{R}^{d}$, we denote the $d\times d$
matrix with the $v_{i}$ as column vectors by $[v_{1},v_{2},\ldots,v_{n}]$. We
remark that since every zonotope $Z$ is centrally symmetric, the quantities
$2\operatorname{cr}(Z)$ and $2\operatorname{ir}(Z)$ are equal to the diameter
and the minimum width of $Z$.

In this paper we distinguish zonotopes satisfying some additional property.
More specifically, if the lengths of all generating vectors of a zonotope $Z$
are equal (resp., equal to $1$), we say that $Z$ is an _equilateral_ (resp.,
_unit edge length_) zonotope. Note that up to translation, any zonotope $Z$ in
$\mathbb{R}^{d}$ can be defined as $Z=\sum_{i=1}^{n}[o,p_{i}]$ for some
$p_{1},p_{2},\ldots,p_{n}\in\mathbb{R}^{d}$. In this case we say that
$Z=\sum_{i=1}^{n}[o,p_{i}]$ is given in a _canonical form_. Clearly, the
zonotope $Z=\sum_{i=1}^{n}[o,p_{i}]$ is symmetric to the origin $o$ if and
only if $\sum_{i=1}^{n}p_{i}=o$; in this case we say that $Z$ has a _centered
canonical form_. We denote the family of zonotopes in $\mathbb{R}^{d}$
generated by at most $n$ segments by $\mathcal{Z}_{d,n}$.

We note that the zonotope $Z=\sum_{i=1}^{n}[o,p_{i}]$,
$p_{i}\in\mathbb{R}^{d}$ is the translate of the zonotope
$Z^{\prime}=\frac{1}{2}\sum_{i=1}^{n}[-p_{i},p_{i}]$. Thus, since $Z^{\prime}$
is $o$-symmetric and every vertex of $Z^{\prime}$ is of the form
$\sum_{i=1}^{n}\varepsilon_{i}p_{i}$ with some $\varepsilon_{i}\in\\{-1,1\\}$,
we have that

(2)
$\operatorname{cr}(Z)=\frac{1}{2}\max\left\\{\left|\sum_{i=1}^{n}\varepsilon_{i}p_{i}\right|:\varepsilon_{i}\in\\{-1,1\\},i=1,2,\ldots,n\right\\}.$

Recall that the support function of a convex body $K\subset\mathbb{R}^{d}$ is
defined as $h_{K}:\mathbb{S}^{d-1}\to\mathbb{R}$, $h_{K}(u)=\sup\\{\langle
x,u\rangle:x\in K\\}$. Furthermore, recall that for any convex bodies
$K,L\subset\mathbb{R}^{d}$, we have $h_{K+L}=h_{K}+h_{L}$, and
$h_{[-q,q]}(u)=|\langle q,u\rangle|$ for any $q\in\mathbb{R}^{n}$. As
$\operatorname{cr}(K)=\max\\{h_{K}(u):u\in\mathbb{S}^{d-1}\\}$ and
$\operatorname{ir}(K)=\min\\{h_{K}(u):u\in\mathbb{S}^{d-1}\\}$, this yields
that for $Z=\sum_{i=1}^{n}[o,p_{i}]$, $p_{i}\in\mathbb{R}^{d}$,

(3) $\operatorname{cr}(Z)=\frac{1}{2}\max\left\\{\sum_{i=1}^{n}|\langle
u,p_{i}\rangle|:u\in\mathbb{S}^{d-1}\right\\},\hbox{ and
}\operatorname{ir}(Z)=\frac{1}{2}\min\left\\{\sum_{i=1}^{n}|\langle
u,p_{i}\rangle|:u\in\mathbb{S}^{d-1}\right\\}.$

Our first goal is to prove a generalization of the decomposition theorem in
[45], which yields an elementary proof of the formula in [15]. To state this
result, we need some preparation.

Let $d\geq 2$, and consider a zonotope $Z=\sum_{i=1}^{n}[o,p_{i}]$ with
$p_{i}\in\mathbb{R}^{d}$ for all values of $i$. We set
$\mathcal{P}^{Z}=\\{p_{1},\ldots,p_{n}\\}$, and for any $0\leq k\leq d$, by
$\mathcal{P}^{Z}_{k}$ the family of $k$-element subsets of $\mathcal{P}^{Z}$
containing linearly independent vectors. Furthermore, for any
$I\in\mathcal{P}^{Z}_{k}$ with $k\geq 1$, we set $P(I)=\sum_{i\in
I}[o,p_{i}]$, and for later use, we extend this notation for the case $k=0$ by
setting $P(\emptyset)=\\{o\\}$, which we regard as a $0$-dimensional
parallelotope. Note that for any
$I=\\{i_{1},i_{2},\ldots,i_{k}\\}\in\mathcal{P}^{Z}_{k}$, the $k$-dimensional
volume of $P(I)$ is $V_{k}(P(I))$, which is equal to the length of the wedge
product of the vectors $p_{i_{j}}$ [15], i.e. $V_{k}(P(I))=|p_{i_{1}}\wedge
p_{i_{2}}\wedge\ldots\wedge p_{i_{k}}|$. In addition, for any parallelotope
$P(I)$ with $I\in\mathcal{P}^{Z}_{k}$, we set
$B^{\perp}(I)=\mathbf{B}^{d}\cap\left(\operatorname{aff}P(I)\right)^{\perp}$
and
$\mathcal{S}^{\perp}(I)=\mathbb{S}^{d-1}\cap\left(\operatorname{aff}P(I)\right)^{\perp}$,
where $\left(\operatorname{aff}P(I)\right)^{\perp}$ denotes the orthogonal
complement of $\operatorname{aff}P(I)$. Finally, by a decomposition of a
convex body $K$ we mean is a finite family of mutually non-overlapping convex
bodies whose union is $K$.

######  Theorem 1.

Using the notation in the previous paragraph, for any $t\geq 0$, the set
$Z+t\mathbf{B}^{d}$ can be decomposed into a family $\mathcal{F}_{Z}$ of
mutually non-overlapping convex bodies of the form $X+tB_{X}$ such that

  1. (1)

for any $X+tB_{X}\in\mathcal{F}_{Z}$, $X$ is a translate of some parallelotope
$P(I)$ with $I\in\mathcal{P}^{Z}_{k}$ for some $0\leq k\leq d$, and
$B_{X}\subseteq B^{\perp}(I)$ is the convex hull of $o$ and a spherically
convex, compact subset of $S^{\perp}(I)$;

  2. (2)

if for any $0\leq k\leq d$ and $I\in\mathcal{P}^{Z}_{k}$, $\mathcal{F}_{Z}(I)$
denotes the subfamily of the elements $X+tB_{X}$ of $\mathcal{F}_{Z}$, where
$X$ is a translate of $P(I)$, then $\\{B_{X}:X+tB_{X}\in\mathcal{F}_{Z}(I)\\}$
is a decomposition of $B^{\perp}(I)$.

Figure 1. The body $Z+t\mathbf{B}^{d}$ if $Z$ is a cube generated by $3$
mutually orthogonal segments. There are $4$ translates of every generating
segment appearing as edges of $Z$. The solid bodies in the picture correspond
to the sets $X+tB_{X}$, where $X$ is a translate of a fixed generating
segment.

We note that the case $t=0$ of Theorem 1 coincides with the decomposition
theorem in [45].

###### Proof.

Recall [45] that a zonotope $Z$ is called _cubical_ if every facet of $Z$ is
an affine cube, or equivalently, if any at most $d$ of its generating vectors
in a canonical form are linearly independent. We prove the statement only for
cubical zonotopes. In particular, we show that in this case Theorem 1 holds
with the additional assumption that if $X+tB_{X}\in\mathcal{F}_{Z}$ with $\dim
X<d$, then $X$ is a face of $Z$ and $B_{X}$ is the set of the outer normal
vectors of $X$ of length at most one, and observe that the general case
follows from this by a standard limit argument.

We prove the theorem by induction on the dimension $d$. Observe that for $d=1$
the statement is trivial. Assume that it holds for any zonotope of dimension
strictly less than $d$, and let $Z=\sum_{i=1}^{n}[o,p_{i}]$,
$p_{i}\in\mathbb{R}^{n}$ be a zonotope. Let $F$ be a $k$-face of $Z$. Then
there is a supporting hyperplane $H$ of $Z$ with $F=H\cap Z$. By the
definition of a zonotope, $F$ is a translate of the generating segments
$[o,p_{i}]$ with the property that $p_{i}$ is parallel to $H$. Thus, since $Z$
is cubical, any $k$-face of $Z$ is a translate of some $P(I)$ with
$I\in\mathcal{P}^{Z}_{k}$. Let
$\operatorname{proj}:\mathbb{R}^{n}\to\left(\operatorname{aff}(P(I))\right)^{\perp}$
be the orthogonal projection onto
$\left(\operatorname{aff}(P(I))\right)^{\perp}$. Then $\operatorname{proj}(Z)$
is a zonotope generated by the projections of the generating segments of $Z$,
and since $Z$ is cubical, the vertices of $\operatorname{proj}(Z)$ coincide
with the projections of the $k$-faces of $Z$ that are translates of $P(I)$.
Since it is well known that the sets of the outer unit normal vectors of a
convex polytope in $\mathbb{R}^{d-k}$ is a decomposition of
$\mathbb{S}^{d-k-1}$ into spherically convex sets (see, e.g. [42]), we proved
that the closure of $(Z+t\mathbf{B}^{d})\setminus Z$ can be decomposed with
sets $X+tB_{X}$, where $X$ is a proper face of $Z$, $B_{X}$ is the set of
outer normal vectors of $X$ of length at most one, and for any
$I\in\mathcal{P}^{Z}_{k}$, $0\leq k<d$,
$\\{B_{X}:X+tB_{X}\in\mathcal{F}_{Z}(I)\\}$ is a decomposition of
$B^{\perp}(I)$.

We are left to show that $Z$ can be decomposed into translates of $P(I)$ with
$I\in\mathcal{P}^{Z}_{d}$ such that for any $I\in\mathcal{P}^{Z}_{d}$, there
is exactly one translate of $P(I)$ in the decomposition. We prove it by
induction on $n$. If $n=d$, the statement is obvious. Assume that it holds for
cubical zonotopes generated by at most $n-1$ segments. Let
$W=\sum_{i=1}^{n-1}[o,p_{i}]$. Then $W$ can be decomposed into translates of
$P(I)$ with $I\in\mathcal{P}^{W}_{d}$ according to the requirements.
Furthermore, by the previous paragraph, for any $J\in\mathcal{P}^{W}_{d-1}$
there are exactly two facets of $W$ and $Z$ that are translates of $P(J)$. By
the definition of a zonotope, if $F_{1}$ and $F_{2}$ are the above two facets
of $W$, then either $F_{1}$ and $p_{n}+F_{2}$, or $F_{2}$ and $p_{n}+F_{1}$
are facets of $Z$. Thus, the closure of $Z\setminus W$ can be decomposed into
translates of $P(J\cup\\{n\\})$, where $J\in\mathcal{P}^{W}_{d-1}$, such that
for each $J\in\mathcal{P}^{W}_{d-1}$, there is exactly one translate of
$P(J\cup\\{n\\})$ in the decomposition. This proves Theorem 1. $\Box$

For Corollary 1, see also [15].

######  Corollary 1.

For any zonotope $Z=\sum_{i=1}^{n}[o,p_{i}]$ in $\mathbb{R}^{d}$, and any
$0\leq k\leq d$, the $k$th intrinsic volume of $Z$ is

(4) $V_{k}(Z)=\sum_{1\leq i_{1}<i_{2}<\ldots<i_{k}\leq n}|p_{i_{1}}\wedge
p_{i_{2}}\wedge\ldots\wedge p_{i_{k}}|.$

###### Proof.

Theorem 1 yields that for any $t>0$

$V_{d}(Z+t\mathbf{B}^{d})=\kappa_{d}t^{d}+\sum_{k=1}^{d}\sum_{1\leq
i_{1}<i_{2}<\ldots<i_{k}\leq n}|p_{i_{1}}\wedge p_{i_{2}}\wedge\ldots\wedge
p_{i_{k}}|\kappa_{d-k}t^{d-k}.$

Corollary 1 readily follows from comparing this formula with Steiner’s formula
in (1). $\Box$

######  Remark 1.

Recall that the mean width $w(K)$ of a convex body $K\subset\mathbb{R}^{d}$ is
$w(K)=\frac{2\kappa_{d-1}}{d\kappa_{d}}V_{1}(K)$ (see [21]). Thus, the mean
width of a zonotope $Z=\sum_{i=1}^{n}[o,p_{i}]$ in $\mathbb{R}^{d}$ is
$w(Z)=\frac{2\kappa_{d-1}}{d\kappa_{d}}\sum_{i=1}^{n}|p_{i}|$.

######  Remark 2.

Let $P$ be a convex polytope in $\mathbb{R}^{d}$ with $m$ facets such that the
outer unit normal vector of the $i$th facet of $P$ is $u_{i}$ and its
$(d-1)$-dimensional volume is $F_{i}$. Recall that for a convex body
$K\in\mathbb{R}^{n}$, the function $h:\mathbb{S}^{d-1}\to\mathbb{R}$,
$h(u)=V_{d-1}(K|x^{\perp})$, where $K|x^{\perp}$ is the orthogonal projection
of $K$ onto the hyperplane through $o$ and with unit normal vector $x$, is the
support function of a convex body; this body, denoted by $\Pi K$, is called
the _projection body_ of $K$ [21]. Cauchy’s projection formula [21] states
that $V_{d-1}(K|x^{\perp})=\int_{\mathbb{S}^{d-1}}|\langle
x,u\rangle|\,dS(K,u)$, where $S(K,\cdot)$ is the surface area measure of $K$.
Applying this for $P$, we obtain that $h_{\Pi
K}(x)=\sum_{i=1}^{m}F_{i}|\langle x,u_{i}\rangle|$ for any
$x\in\mathbb{S}^{d-1}$. Thus, the additivity of the support function with
respect to Minkowski sums of convex bodies yields that

$\Pi P=\sum_{i=1}^{m}[o,F_{i}u_{i}]$

In particular, by Corollary 1 and since both mean width and surface area are
continuous functions with respect to Hausdorff distance, for any convex body
$K\subset\mathbb{R}^{d}$, we have $w(\Pi
K)=\frac{2\kappa_{d-1}}{d\kappa_{d}}\operatorname{surf}(K)$.

In the remaining part of Section 2, we introduce two problems that are related
to isoperimetric problems for zonotopes.

For $p>0$ and a multiset $\omega_{n}=\\{x_{1},x_{2},\ldots,x_{n}\\}$ in
$\mathbb{S}^{d-1}$, the _$\ell_{p}$ -polarization of $\omega_{n}$_ is defined
as

$M_{p}(\omega_{n})=\max\left\\{\sum_{i=1}^{n}\left|\langle
x_{i},u\rangle\right|^{p}:u\in\mathbb{S}^{d-1}\right\\},$

and the quantity

$M_{n}^{p}(\mathbb{S}^{d-1})=\min\left\\{M_{p}(\omega_{n}):\omega_{n}\subset\mathbb{S}^{d-1}\right\\}$

is called the _$\ell_{p}$ -polarization (or Chebyshev) constant of
$\mathbb{S}^{d-1}$_. The $\ell_{p}$-polarization problem on the sphere asks
for determining the value of $M_{n}^{p}(\mathbb{S}^{d-1})$ for all values of
$p$ and $d$.

Note that by (2) and (3), if $Z=\sum_{i=1}^{n}[o,x_{i}]$ is a zonotope in
$\mathbb{R}^{d}$, then

(5)
$\operatorname{cr}(Z)=\frac{1}{2}\max\left\\{\left|\sum_{i=1}^{n}\varepsilon_{i}x_{i}\right|:\varepsilon_{i}\in\\{-1,1\\},i=1,2,\ldots,n\right\\},$

or equivalently (using the definition of support function),

(6) $\operatorname{cr}(Z)=\frac{1}{2}\max\left\\{\sum_{i=1}^{n}|\langle
u,x_{i}\rangle|:u\in\mathbb{S}^{d-1}\right\\}.$

Thus, the $\ell_{1}$-polarization constant of $\mathbb{S}^{d-1}$ is equal to
twice the minimal circumradius of a zonotope generated by $n$ segments of unit
length. By Remark 1, the mean width of a zonotope is a scalar multiple of the
total length of its generating vectors. Thus, this problem can be restated as
finding the minimum circumradius of a zonotope generated by $n$ segments of
unit length. In particular, for the special case of unit vectors, the equality
of the expressions in (5) and (6) is proved as Proposition 3 of [3] by the
Lagrange multiplier method.

The function $\sigma^{k}_{m}(x_{1},\ldots,x_{m})=\sum_{1\leq
i_{1}<i_{2}<\ldots<i_{k}\leq m}x_{i_{1}}x_{i_{2}}\ldots x_{i_{k}}$ is called
the $k$th elementary symmetric function on the $m$ variables
$x_{1},\ldots,x_{m}$. The following statement, called Maclaurin’s inequality,
can be found e.g. in [10].

######  Lemma 1 (Maclaurin’s inequality).

Let $1\leq k<m$ be integers, and $x_{1},\ldots,x_{m}>0$ be positive real
numbers. Then

$\left(\frac{\sigma_{m}^{k}(x_{1},x_{2},\ldots,x_{m})}{\binom{m}{k}}\right)^{\frac{1}{k}}\geq\left(\frac{\sigma_{m}^{\textbf{k+1}}(x_{1},x_{2},\ldots,x_{m})}{\binom{m}{k+1}}\right)^{\frac{1}{k+1}},$

with equality if and only if all $x_{i}$ are equal.

A generalization of this inequality for a set of vectors in $\mathbb{R}^{d}$
was conjectured in [15] as follows.

######  Conjecture 1 (Brazitikos, McIntyre).

Let $x_{1},x_{2},\ldots,x_{n}\in\mathbb{R}^{d}$ be given with $1\leq d\leq n$.
Then for any $p\in[0,\infty]$ and $2\leq k\leq d$, we have

(7) $\left(\frac{\sum_{1\leq i_{1}<\ldots<i_{k}\leq n}|x_{i_{1}}\wedge
x_{i_{2}}\wedge\ldots\wedge
x_{i_{k}}|^{p}}{\binom{n}{k}}\right)^{\frac{1}{pk}}\leq\left(\frac{\sum_{1\leq
i_{1}<\ldots<i_{k-1}\leq n}|x_{i_{1}}\wedge x_{i_{2}}\wedge\ldots\wedge
x_{i_{k-1}}|^{p}}{\binom{n}{k-1}}\right)^{\frac{1}{p(k-1)}},$

with equality if and only if $n=d$ and the vectors form an orthonormal basis.

The authors of [15] proved this conjecture for $p=0$ and $p=\infty$, for $p=2$
and $n=d$, and for $p=1$, $n=d$ and $k=2,3,d$. They also pointed out (see also
Corollary 1) that if $p=1$, the numerators in (7) correspond to the $k$th and
$(k-1)$st intrinsic volumes of the zonotope $\sum_{i=1}^{n}[o,x_{i}]$, showing
that this case of Conjecture 1 directly leads to isoperimetric problems for
zonotopes.

##  3\. Isoperimetric problems for zonotopes generated by a small number of
segments

Zonotopes in $\mathcal{Z}_{d,d}$ are called parallelotopes, and those in
$\mathcal{Z}_{d,d+1}$ are called rhombic dodecahedra [9]. The goal of this
section is to prove isoperimetric inequalities for them. We note that if
$Z=\sum_{i=1}^{d+1}[o,p_{i}]$ is a rhombic dodecahedron where the points
$p_{i}$ are the vertices of a regular simplex centered at $o$, then $Z$ is
called a _regular rhombic dodecahedron_. A regular rhombic dodecahedron in
$\mathbb{R}^{3}$ is the Voronoi cell of a face-centered cubic lattice. Thus,
by the seminal result of Hales [27] proving Kepler’s conjecture, among rhombic
dodecahedra in $\mathbb{R}^{3}$ with unit inradius, the ones with minimal
volume are the regular ones. This consequence of the result of Hales was
strengthened in [9], which we quote for completeness.

######  Theorem 2 (Bezdek).

Let $1\leq k\leq d$ be arbitrary. Among rhombic dodecahedra in
$\mathbb{R}^{d}$ of unit inradius, the ones with minimal $k$th intrinsic
volumes are the regular ones.

Intuitively, regular rhombic dodecahedra appear as natural candidates for
solutions in isoperimetric problems for rhombic dodecahedra. Nevertheless, as
we will see in Subsection 3.2, this does not always hold.

In Subsection 3.1 we find the minimal intrinsic volumes and circumradii of the
elements of $\mathcal{Z}_{d,d}$ and $\mathcal{Z}_{d,d+1}$ with a given volume.
In Subsections 3.2 and 3.3 we investigate the problems of finding the minimal
circumradius and the second intrinsic volume $V_{2}(\cdot)$, respectively, of
a parallelotope or a rhombic dodecahedron of a given mean width, respectively.

In Subsection 3.4 our investigation is motivated by a result of Tanner [46],
who defined the _total squared $k$-content of a simplex
$S\subset\mathbb{R}^{n}$_ as the sum of the squares of the $k$-volumes of its
$k$-faces, and proved that among simplices with a given squared $1$-content,
regular simplices have maximal squared $k$-content for all $1\leq k\leq d$. In
this subsection we define the total squared $k$-volume of a zonotope with
$1\leq k\leq d$, and find its minimum among the parallelotopes and rhombic
dodecahedra with a given squared $l$-volume for all $k<l\leq d$.

In our investigation, we set $\mathcal{Z}_{\mathrm{p}}=\mathcal{Z}_{d,d}$ and
$\mathcal{Z}_{\mathrm{rd}}=\mathcal{Z}_{d,d+1}$, and we denote by
$Z_{\mathrm{p}}^{\mathrm{reg}}\in\mathcal{Z}_{d,d}$ a cube, and by
$Z_{\mathrm{rd}}^{\mathrm{reg}}\in\mathcal{Z}_{d,d+1}$ a regular rhombic
dodecahedron.

###  3.1. The intrinsic volumes and circumradius of a zonotope with a given
volume

Our main result in Subsection 3.1 is Theorem 3.

######  Theorem 3.

Let $1\leq k\leq d-1$. Then, for any $Z_{i}\in\mathcal{Z}_{i}$ with
$V_{d}(Z_{i})=V_{d}(Z_{i}^{\mathrm{reg}})$, where
$i\in\\{\mathrm{p},\mathrm{rd}\\}$, we have

$V_{k}(Z_{i})\geq V_{k}(Z_{i}^{\mathrm{reg}})$

with equality if and only if $Z_{i}$ is congruent to $Z_{i}^{\mathrm{reg}}$.
Furthermore, we have

$\operatorname{cr}(Z_{i})\geq\operatorname{cr}(Z_{i}^{\mathrm{reg}}).$

We note that our next corollary readily follows from Theorems 2 and 3.

######  Corollary 2.

For any $Z_{i}\in\mathcal{Z}_{i}$ with
$\operatorname{ir}(Z_{i})=\operatorname{ir}(Z_{i}^{\mathrm{reg}})$, where
$i\in\\{\mathrm{p},\mathrm{rd}\\}$, we have

$\operatorname{cr}(Z_{i})\geq\operatorname{cr}(Z_{i}^{\mathrm{reg}})$

with equality if and only if $Z_{i}$ is congruent to $Z_{i}^{\mathrm{reg}}$.

We prove Theorem 3 only for $i=\mathrm{rd}$, as the same argument can be used
to prove it for $i=\mathrm{p}$. We note that the case $i=\mathrm{p}$ and
$k=d-1$ of our theorem was also proved in Theorem 5 of [15]. Our proof is
based on an application of shadow systems [41] (also called linear parameter
systems [40]) and Steiner symmetrization. We start with two lemmas needed for
the proof.

######  Lemma 2.

Let $k\geq 2$, $p_{1},\ldots,p_{k},v\in\mathbb{R}^{d}$, and
$\lambda_{1},\ldots,\lambda_{k}\in\mathbb{R}$. For $i=1,2,\ldots,k$ and all
$t\in\mathbb{R}$, set $p_{i}(t)=p_{i}+\lambda_{i}tv$ and
$S(t)=\operatorname{conv}\\{p_{1}(t),\ldots,p_{k}(t)\\}$. Then the function
$f:\mathbb{R}\to\mathbb{R}$, $f(t)=\operatorname{vol}_{k-1}(S(t))$ is a convex
function of $t$, and if the points $p_{1},\ldots,p_{k},v$ are affinely
independent, then $f$ is strictly convex.

###### Proof.

Since $\bigcup_{t\in\mathbb{R}}S(t)$ is contained in a $k$-dimensional affine
subspace of $\mathbb{R}^{d}$, without loss of generality, we may assume that
$k=d$. Furthermore, since volume is invariant under translations,
$V_{d}(S(t))=V_{d}(S(t)-p_{d}(t))$, and thus, we may also assume that
$p_{d}=o$ and $\lambda_{d}=0$. Define the linear functional
$L_{t}:\mathbb{R}^{d}\to\mathbb{R}$,
$L_{t}(x)=\det([p_{1}(t),\ldots,p_{d-1}(t),x])$. This functional can be
written in the form $L_{t}(x)=\langle u(t),x\rangle$, where $u(t)$ is the
vector whose entries are the (signed) minors of size $(d-1)\times(d-1)$ of the
$d\times(d-1)$ matrix $[p_{1}(t),\ldots,p_{d-1}(t)]$. Note that by the
properties of determinants, $u(t)$ is a linear function of $t$, and hence, it
can be written as $u(t)=u+tw$ for some $u,w\in\mathbb{R}^{n}$. Furthermore,
$f(t)=\operatorname{vol}_{d-1}(S(t))=\frac{1}{(d-1)!}|u(t)|$. Thus,

$f^{\prime\prime}(t)=\frac{\langle w,w\rangle\langle u(t),u(t)\rangle-\langle
w,u(t)\rangle^{2}}{(d-1)!|u(t)|^{3}}\geq 0$

by the Cauchy-Schwartz Inequality. This implies the convexity of $f$.

Now, assume that $p_{1},\ldots,p_{k},v$ are affinely independent. Then, as
$p_{d}=o$, it follows that $p_{1},\ldots,p_{d-1},v$ are linearly independent.
Note that $u(t)\neq o$ for any $t\in\mathbb{R}$. Indeed, if $u(t)=o$ for some
value of $t$, then, by the definition of $L_{t}(x)$,
$p_{1}(t),\ldots,p_{d-1}(t)$ are linearly dependent for some value of $t$,
which yields the existence of a nontrivial linear combination of
$p_{1},\ldots,p_{d-1},v$ equal to $o$. Thus, we have $u(t)\neq o$ for any
$t\in\mathbb{R}$, implying that $u$ is not a scalar multiple of $w$. We show
that $w\neq o$. Indeed, suppose for contradiction that $w=o$. Then the value
of $L_{t}(x)$ is independent of $t$ for any value of $x$. In particular, the
kernel $\operatorname{Ker}1(L_{t})$ of $L_{t}$ is independent of $t$, implying
that $p_{1},\ldots,p_{d-1},v\in\operatorname{Ker}1(L_{t})$. Thus, combined
with the fact that $L_{t}$ is not identically zero, this yields that
$p_{1},\ldots,p_{d-1},v$ are linearly dependent, contradicting our
assumptions. Now, we have shown that $w\neq o$, from which we also have that
$u(t)$ and $w$ are not scalar multiples of each other for any value of $t$.
From this, the Cauchy-Schwartz Inequality implies that $f^{\prime\prime}(t)>0$
for any value of $t$, and hence $f$ is strictly convex. $\Box$

######  Lemma 3.

For any simplex $S\subset\mathbb{R}^{d}$ and $1\leq k\leq d-1$, let
$\mathcal{F}_{k}(S)$ denote the family of the $(k-1)$-faces of $S$, and set
$f_{k}(S)=\sum_{F\in\mathcal{F}_{k}(S)}V_{k}(\\{o\\}\cup F)$. Furthermore, if
$S=\operatorname{conv}\\{p_{1},p_{2},\ldots,p_{d+1}\\}$, set

$g(S)=\max\left\\{\left|\sum_{i=1}^{d+1}\varepsilon_{i}p_{i}\right|:\varepsilon_{i}\in\\{-1,1\\},i=1,2,\ldots,d+1\right\\}.$

Let $S_{\mathrm{reg}}$ be a regular simplex centered at $o$. Then, for any
$1\leq k\leq d-1$ and any simplex $S\subset\mathbb{R}^{d}$ with
$V_{d}(S)=V_{d}(S_{\mathrm{reg}})$, we have

$f_{k}(S)\geq f_{k}(S_{\mathrm{reg}})$

with equality if and only if $S$ is a regular simplex centered at $o$.
Furthermore, we also have

$g(S)\geq g(S_{\mathrm{reg}}).$

###### Proof.

Without loss of generality, let $V_{d}(S_{\mathrm{reg}})=1$. Clearly, the
function $f_{k}(\cdot)$ attains its minimum value in the family of simplices
with unit volume. Thus, it is sufficient to show that if $f_{k}$ is minimal at
a simplex $S$ with $V_{d}(S)=1$, then it is a regular simplex centered at $o$.

Consider a simplex $S$ with $V_{d}(S)=1$, which is not a regular simplex
centered at $o$. Let $S=\operatorname{conv}\\{p_{1},p_{2},\ldots,p_{d+1}\\}$,
and let $H$ be the unique hyperplane perpendicular to the line through the
edge $E=[p_{1},p_{2}]$ and passing through $o$.

Figure 2. Lemma 2 in case $d=3$.

Without loss of generality, assume that $S$ is not symmetric to $H$, and let
$S^{\prime}$ denote the reflected copy of $S$ about $H$. Let $v$ be a unit
normal vector of $H$ and let $\delta_{i}$ denote the signed distance of
$p_{i}$ from $H$, where the sign is chosen in such a way that distances are
positive in the open half space bounded by $H$ and containing $v$. Note that
for any value of $i$, $p_{i}-\delta_{i}v$ is the orthogonal projection of
$p_{i}$ onto $H$, which we denote by $q_{i}$, and set $q=q_{1}=q_{2}$. For any
$i\geq 3$, let $p_{i}(t)=q_{i}+\delta_{i}tv$, and let
$p_{1}(t)=q+\frac{\delta_{1}-\delta_{2}}{2}v+\frac{\delta_{1}+\delta_{2}}{2}tv$
and
$p_{2}(t)=q+\frac{\delta_{2}-\delta_{1}}{2}v+\frac{\delta_{1}+\delta_{2}}{2}tv$.
Finally, let $S(t)=\operatorname{conv}\\{p_{1}(t),\ldots,p_{d+1}(t)\\}$.
Clearly, $S(1)=S$, $S(-1)=S^{\prime}$, and $S(0)$ is the Steiner symmetral of
$S$ with respect to $H$, and hence, it is symmetric to $H$. By Lemma 2,
$t\mapsto f_{k}(S(t))$ is a strictly convex even function on $[-1,1]$, and
hence, it attains a unique minimum at $t=0$. This yields the assertion for
$f_{k}(S)$.

Finally, we consider the function $g$. Let $\mathcal{F}_{\min}$ denote the
subfamily of the family of unit volume simplices that minimize $g$ in this
family. Recall that $f_{1}(S)=\sum_{i=1}^{d+1}|p_{i}|$. Note that $f_{1}$
attains its minimal value in $\mathcal{F}_{\min}$. Thus, to prove the
assertion it is sufficient to show that if $S$ is not a unit volume regular
simplex centered at $o$, then there is a unit volume simplex $S^{\prime}$ such
that $g(S^{\prime})<g(S)$, or $g(S^{\prime})=g(S)$ and
$f_{1}(S^{\prime})<f_{1}(S)$. To do it we use the notation of the previous
paragraph. Note that the function
$t\mapsto|\sum_{i=1}^{d+1}\varepsilon_{i}p_{i}(t)|$ is convex for any
$(\varepsilon_{1},\ldots,\varepsilon_{d+1})\in\\{-1,1,\\}^{d+1}$. Since the
maximum of convex functions is convex, $g(S(t))$ is an even convex function on
$[-1,1]$, implying that $g(S(0))\leq g(S(1))$. Clearly, since $S$ is not
symmetric to $H$, we also have that $f_{1}(S(0))<f_{1}(S(1))$, which yields
the statement. $\Box$

Now we prove Theorem 3.

###### Proof of Theorem 3.

Let $Z_{\mathrm{rd}}^{\mathrm{reg}}=\sum_{i=1}^{d+1}[o,q_{i}]$, and set
$S_{\mathrm{reg}}=\operatorname{conv}\\{q_{1},q_{2},\ldots,q_{d+1}\\}$. Then
$S_{\mathrm{reg}}$ is a regular simplex centered at $o$, and by Corollary 1,
$V_{d}(S_{\mathrm{reg}})=\frac{1}{d!}{\color[rgb]{0,0,0}V_{d}(Z_{\mathrm{rd}}^{\mathrm{reg}})}$.
Now, consider any rhombic dodecahedron $Z$ with
$V_{d}(Z)=V_{d}(Z_{\mathrm{rd}}^{\mathrm{reg}})$. Since the statement in
Theorem 3 is invariant under translating $Z$, we may assume that
$Z=\sum_{i=1}^{d+1}[o,p_{i}]$, where $o\in
S=\operatorname{conv}\\{p_{1},p_{2},\ldots,p_{d+1}\\}$. Clearly, this implies
that $S$ is a nondegenerate simplex with volume $V_{d}(S_{\mathrm{reg}})$.
Thus, applying Lemma 3 for $S$ and $S_{\mathrm{reg}}$ readily implies Theorem
3. $\Box$

By Theorem 3 and the convexity of the function $x\mapsto x^{p}$, where $p>1$,
we readily obtain the following.

######  Remark 3.

Let $x_{1},x_{2},\ldots,x_{n}\in\mathbb{R}^{d}$ with $d\leq n\leq d+1$. Then
for all $1\leq k\leq d$,

$\left(\frac{\sum_{1\leq i_{1}<\ldots<i_{k}\leq n}|x_{i_{1}}\wedge
x_{i_{2}}\wedge\ldots\wedge
x_{i_{k}}|}{\binom{n}{k}}\right)^{\frac{1}{k}}\geq\left(\frac{\sum_{1\leq
i_{1}<\ldots<i_{d}\leq n}|x_{i_{1}}\wedge x_{i_{2}}\wedge\ldots\wedge
x_{i_{d}}|}{\binom{n}{d}}\right)^{\frac{1}{d}},$

with equality if and only if $n=d$ and the vectors are pairwise orthogonal and
are of equal length, or if the linear hull of the $x_{i}$ is of dimension at
most $k-1$. Furthermore, for any $p>1$, we have

$\left(\frac{\sum_{1\leq i_{1}<\ldots<i_{k}\leq d}|x_{i_{1}}\wedge
x_{i_{2}}\wedge\ldots\wedge
x_{i_{k}}|^{p}}{\binom{n}{k}}\right)^{\frac{1}{pk}}\geq|x_{1}\wedge
x_{2}\wedge\ldots\wedge x_{d}|^{\frac{1}{d}},$

with equality if and only if the vectors are pairwise orthogonal and are of
equal length, or if the linear hull of the $x_{i}$ is of dimension at most
$k-1$.

For any simplex $S\subset\mathbb{R}^{d}$, integer $1\leq k\leq d-1$, and real
number $m\in\mathbb{R}$, let $g_{k}^{m}(S)$ denote the sum of the $m$th powers
of the volumes of the $k$-faces of $S$. Tanner [46] showed that if $2\leq
k\leq d-1$ and $m\in(0,2]$, among simplices $S$ in $\mathbb{R}^{n}$ with
$g_{1}^{2}(S)$ fixed, $g_{k}^{m}(S)$ is maximal if and only if $S$ is regular.
Our last result in this subsection is Proposition 3.1, which can be regarded
as a complement of the above result of Tanner. This result follows by a
natural modification of the proof of Lemma 3, and the observation that the
composition of convex functions is convex.

######  Proposition 3.1.

Let $1\leq k\leq d-1$ be an integer, and $m\geq 1$ be a real number. Then, for
any simplex $S\subset\mathbb{R}^{d}$ with unit volume, $g_{k}^{m}(S)$ is
maximal if and only if $S$ is regular.

###  3.2. The circumradius of a zonotope with a given mean width

Our aim in this section is to find the parallelotopes or rhombic dodecahedra
$Z$ of a given mean width $w(Z)$ that have minimal circumradius. For
parallelotopes, we have the following theorem.

######  Theorem 4.

If $Z\in\mathcal{Z}_{\mathrm{p}}$ satisfies
$w(Z)=w(Z_{\mathrm{p}}^{\mathrm{reg}})$, then
$\operatorname{cr}(Z)\geq\operatorname{cr}(Z_{\mathrm{p}}^{\mathrm{reg}})$,
with equality if and only if $Z$ is a cube.

###### Proof.

By (2),

$4\cdot
2^{d}\cdot\left(\operatorname{cr}(Z)\right)^{2}\geq\sum_{(\varepsilon_{1},\ldots,\varepsilon_{d})\in\\{-1,1\\}^{d}}\left|\sum_{i=1}^{d}\varepsilon_{i}p_{i}\right|^{2}=2^{d}\sum_{i=1}^{d}|p_{i}|^{2},$

with equality if and only if the vectors $p_{i}$ are pairwise orthogonal. On
the other hand,

$\sqrt{\frac{\sum_{i=1}^{d}|p_{i}|^{2}}{d}}\geq\frac{\sum_{i=1}^{d}|p_{i}|}{d},$

with equality if and only if all $p_{i}$ are of equal length. By Remark 1,
this yields Theorem 4. $\Box$

Next, we examine $\mathcal{Z}_{\mathrm{rd}}$. We prove that in the family of
rhombic dodecahedra having a centered canonical form, the ones with a given
mean width and minimal circumradius are regular, whereas in the family of all
rhombic dodecahedra this statement is not true. Recall that a zonotope
$\sum_{i=1}^{n}[o,p_{i}]$ is given in a centered canonical form if
$\sum_{i=1}^{n}p_{i}=o$.

We start with a simple observation.

######  Remark 4.

Let $Z_{\mathrm{rd}}^{\mathrm{reg}}=\sum_{i=1}^{d+1}[o,q_{i}]$ with
$q_{i}\in\mathbb{S}^{d-1}$ for all values of $i$. Then

$\operatorname{cr}(Z_{\mathrm{rd}}^{\mathrm{reg}})=\left\\{\begin{array}[]{cc}\frac{\sqrt{d+2}}{2}&\text{if
$d$ is even,}\\\ \frac{d+1}{2\sqrt{d}}&\text{if $d$ is odd.}\\\
\end{array}\right.$

Furthermore, for some $\varepsilon_{i}\in\\{-1,1\\}$, $i=1,2,\ldots,d+1$ we
have
$\frac{1}{2}|\sum_{i=1}\varepsilon_{i}q_{i}|=\operatorname{cr}(Z_{\mathrm{rd}}^{\mathrm{reg}})$
if and only if the number of the $\varepsilon_{i}$ equal to $-1$ is
$\lfloor\frac{d+1}{2}\rfloor$ or $\lceil\frac{d+1}{2}\rceil$.

###### Proof.

Clearly, the value of $|\sum_{i=1}^{d+1}\varepsilon_{i}p_{i}|$ depends only on
the numbers of the positive and negative coefficients. Thus, we may assume
that there is some $0<k\leq d+1$ such that
$\varepsilon_{1}=\ldots=\varepsilon_{k}=1$ and
$\varepsilon_{k+1}=\ldots=\varepsilon_{d+1}=-1$. Furthermore, observe that
$\sum_{i=1}^{d+1}p_{i}=o$ implies that
$\frac{1}{2}|\sum_{i=1}^{k}p_{i}-\sum_{i=k+1}^{d+1}p_{i}|=|\sum_{i=1}^{k}p_{i}|$.
By the same fact and since the value of $\langle p_{i},p_{j}\rangle$ is
independent of $i,j$ for all $i\neq j$, we also have $\langle
p_{i},p_{j}\rangle=-\frac{1}{d}$ for any $i\neq j$. Thus, an elementary
computation yields that

$|p_{1}+\ldots+p_{k}|^{2}=\langle
p_{1}+\ldots+p_{k},p_{1}+\ldots+p_{k}\rangle=k+2\binom{k}{2}\left(-\frac{1}{d}\right)=\frac{k(d+1-k)}{d}.$

As $k$ is an integer, from this we have that the maximum of
$|p_{1}+\ldots+p_{k}|$ is attained if and only if
$k=\lfloor\frac{d+1}{2}\rfloor$ or $k=\lceil\frac{d+1}{2}\rceil$. $\Box$

We note that Theorem 5 in the special case that $d$ is even and all $p_{i}$
are unit vectors is proved in [2, Theorem 6] based on a different idea.

######  Theorem 5.

Let $d\geq 2$ and let
$Z_{\mathrm{rd}}^{\mathrm{reg}}=\sum_{i=1}^{d+1}[o,q_{i}]$. Then for any
$Z=\sum_{i=1}^{d+1}[o,p_{i}]\in\mathcal{Z}_{\mathrm{rd}}$ having a centered
canonical form with $w(Z)=w(Z_{\mathrm{rd}}^{\mathrm{reg}})$, we have

(8)
$\operatorname{cr}(Z)\geq\operatorname{cr}(Z_{\mathrm{rd}}^{\mathrm{reg}}),$

with equality if and only if $Z$ is congruent to
$Z_{\mathrm{rd}}^{\mathrm{reg}}$. Furthermore, if $d$ is odd, then there is a
rhombic dodecahedron $Z^{\prime}=\sum_{i=1}^{d+1}[o,p^{\prime}_{i}]$ with
$w(Z^{\prime})=w(Z_{\mathrm{rd}}^{\mathrm{reg}})$ and
$\operatorname{cr}(Z^{\prime})<\operatorname{cr}(Z_{\mathrm{rd}}^{\mathrm{reg}})$.

###### Proof.

Let $Z=\sum_{i=1}^{d+1}[o,p_{i}]$ with
$w(Z)=w(Z_{\mathrm{rd}}^{\mathrm{reg}})$. Without loss of generality, we
assume that $\sum_{i=1}^{d+1}|q_{i}|=d+1$, which by Remark 1 implies that
$\sum_{i=1}^{d+1}|p_{i}|=d+1$. First, we prove the inequality
$\operatorname{cr}(Z)\geq\operatorname{cr}(Z_{\mathrm{rd}}^{\mathrm{reg}})$
for any $Z\in\mathcal{Z}_{\mathrm{rd}}$ with
$w(Z)=w(Z_{\mathrm{rd}}^{\mathrm{reg}})$ and having a centered canonical form.
To do it, we prove an inequality valid for any rhombic dodecahedron with
$w(Z)=w(Z_{\mathrm{rd}}^{\mathrm{reg}})$. On the other hand, for simplicity we
derive this formula only in the special case that $d$ is odd; if $d$ is even,
the same argument can be applied with a slightly different computation. In the
proof, we denote the coordinates of any $\varepsilon\in\\{-1,1\\}^{d+1}$ by
$\varepsilon=(\varepsilon_{1},\ldots,\varepsilon_{d+1})$, and set
$\mathcal{E}=\left\\{\varepsilon\in\\{-1,1\\}^{d+1}:\sum_{i=1}^{d+1}\varepsilon_{i}=0\right\\}$.

Assume that $d=2m-1$. Then we clearly have

(9)
$2\operatorname{cr}(Z)\geq\max\left\\{\left|\sum_{i=1}^{d+1}\varepsilon_{i}p_{i}\right|:\varepsilon\in\mathcal{E}\right\\}\geq\sqrt{\frac{1}{\binom{d+1}{m}}\sum_{\varepsilon\in\mathcal{E}}\left|\sum_{i=1}^{d+1}\varepsilon_{i}p_{i}\right|^{2}}.$

Furthermore,

(10)
$\sum_{\varepsilon\in\mathcal{E}}\left|\sum_{i=1}^{d+1}\varepsilon_{i}p_{i}\right|^{2}=\sum_{\varepsilon\in\mathcal{E}}\left\langle\sum_{i=1}^{d+1}\varepsilon_{i}p_{i},\sum_{i=1}^{d+1}\varepsilon_{i}p_{i}\right\rangle=\binom{d+1}{m}\sum_{i=1}^{d+1}|p_{i}|^{2}+2\sum_{\varepsilon\in\mathcal{E}}\sum_{1\leq
i<j\leq d+1}\varepsilon_{i}\varepsilon_{j}\langle p_{i},p_{j}\rangle.$

An elementary computation shows that for any $i<j$, the number of the elements
$\varepsilon\in\mathcal{E}$ with the property that
$\varepsilon_{i}=\varepsilon_{j}$ is $2\binom{2m-2}{m-2}$ and the number of
elements with $\varepsilon_{i}=-\varepsilon_{j}$ is $2\binom{2m-2}{m-1}$. As
$2\binom{2m-2}{m-2}-2\binom{2m-2}{m-1}=-\frac{1}{2m-1}\binom{2m}{m}=-\frac{1}{d}\binom{d+1}{m}$,
this implies that

(11) $\sum_{\varepsilon\in\mathcal{E}}\sum_{1\leq i<j\leq
d+1}\varepsilon_{i}\varepsilon_{j}\langle
p_{i},p_{j}\rangle=-\frac{1}{d}\binom{d+1}{m}\sum_{1\leq i<j\leq d+1}\langle
p_{i},p_{j}\rangle.$

On the other hand,

$\sum_{1\leq i<j\leq d+1}\langle
p_{i},p_{j}\rangle=\frac{1}{2}\left(\left\langle\sum_{i=1}^{d+1}p_{i},\sum_{j=1}^{d+1}p_{j}\right\rangle-\sum_{i=1}^{d+1}|p_{i}|^{2}\right)=\frac{1}{2}\left(|p|^{2}-\sum_{i=1}^{d+1}|p_{i}|^{2}\right),$

where $p=\sum_{i=1}^{d+1}p_{i}$. Substituting it back into (11) and applying
(10) and (9), we obtain that

$2\operatorname{cr}(Z)\geq\sqrt{\frac{d+1}{d}\sum_{i=1}^{d+1}|p_{i}|^{2}-\frac{1}{d}|p|^{2}}.$

Now, if $Z$ is given in a centered canonical form, then $p=o$, which implies,
together with the inequality between the arithmetic and the quadratic means,
that

$\operatorname{cr}(Z)\geq\frac{1}{2}\sqrt{\frac{d+1}{d}\sum_{i=1}^{d+1}|p_{i}|^{2}}\geq\frac{d+1}{2\sqrt{d}}\cdot\frac{\sum_{i=1}^{d+1}|p_{i}|}{d+1}\geq\frac{d+1}{2\sqrt{d}}.$

This, combined with Remark 4, yields the inequality in (4). Assume that we
have equality in (4). Then, by the inequality between the arithmetic and the
quadratic means, $|p_{i}|=1$ for all values of $i$. Furthermore, by (11), for
any $\varepsilon\in\mathcal{E}$ we have
$\left|\sum_{i=1}^{d+1}\varepsilon_{i}p_{i}\right|=\frac{d+1}{\sqrt{d}}$, or
equivalently, for any subset $J\subset\\{1,2,\ldots,d+1\\}$ consisting of $m$
indices, we have that $\left|\sum_{j\in J}p_{j}\right|=\frac{d+1}{2\sqrt{d}}$.

Consider an arbitrary $J\subset\\{1,2,\ldots,d+1\\}$ such that $|J|=m-1$ and
$1,2,\notin J$. Then, by the previous observation, $\left|p_{1}+\sum_{j\in
J}p_{j}\right|^{2}=\left|p_{2}+\sum_{j\in J}p_{j}\right|^{2}$. Computing both
sides and using the fact that $|p_{1}|=|p_{2}|=1$, we obtain that
$\left\langle p_{1},\sum_{j\in J}p_{j}\right\rangle=\left\langle
p_{2},\sum_{j\in J}p_{j}\right\rangle$, which yields also that
$\left|p_{1}-\sum_{j\in J}p_{j}\right|=\left|p_{2}-\sum_{j\in J}p_{j}\right|$.
Thus, $\sum_{j\in J}p_{j}$ lies in the bisector $H_{12}$ of the segment
$[p_{1},p_{2}]$. Note that $o\in H_{12}$ follows from the fact that
$|p_{1}|=|p_{2}|$. Observe that our conditions imply that for any $i,j\geq 3$,
we have $p_{i}-p_{j}\in H_{12}$. Thus, if some $p_{j}$ with $j\geq 3$, say
$p_{3}$ does not lie in $H_{12}$, then all $p_{j}$ lie in the hyperplane
$p_{3}+H_{12}$. This yields that for any $J$ satisfying the conditions,
$\sum_{j\in J}p_{j}\in(m-1)p_{3}+H_{12}\neq H_{12}$, a contradiction. Hence,
we have that $p_{j}\in H_{12}$ for all $j\geq 3$, implying that
$\operatorname{conv}\\{p_{1},p_{2},\ldots,p_{d+1}\\}$ is symmetric to the
hyperplane $H_{12}$. Repeating the same argument for any pair of indices in
place of $1$ and $2$, we have that
$S=\operatorname{conv}\\{p_{1},p_{2},\ldots,p_{d+1}\\}$ is symmetric to any
hyperplane through $o$ and a $(d-2)$-face of $S$. This yields that $S$ is a
regular simplex centered at $o$, or in other words, $Z$ is a regular rhombic
dodecahedron.

Now we show that for $d$ odd, if we drop the condition that $Z$ is centered,
then, among rhombic dodecahedra with a given mean width, non-regular ones have
minimal circumradius. Consider the regular rhombic dodecahedron
$Z_{\mathrm{rd}}^{\mathrm{reg}}=\sum_{i=1}^{d+1}[o,q_{i}]$, and let $v\neq o$
be an arbitrary vector. Let $\bar{Z}=\sum_{i=1}^{d+1}[o,q_{i}+v]$. By Remark
4, if $|v|$ is sufficiently small, then

$\operatorname{cr}(\bar{Z})=\left|\sum_{i=1}^{d+1}\varepsilon_{i}(q_{i}+v)\right|,\varepsilon\in\\{-1,1\\}^{d+1}$

implies that $\sum_{i=1}^{d+1}\varepsilon_{i}=0$. As $d$ is odd, from this we
have that
$\sum_{i=1}^{d+1}\varepsilon_{i}(q_{i}+v)=\sum_{i=1}^{d+1}\varepsilon_{i}q_{i}$,
which yields that
$\operatorname{cr}(\bar{Z})=\operatorname{cr}(Z_{\mathrm{reg}})$.

We show that $w(\bar{Z})>w(Z_{\mathrm{rd}}^{\mathrm{reg}})$. To do it it is
sufficient to show that the unique point $x\in
S_{\mathrm{reg}}=\operatorname{conv}\\{q_{1},q_{2},\ldots,q_{d+1}\\}$ for
which the function $\sum_{i=1}^{d+1}|x-q_{i}|$ is minimal is $o$. To prove it,
let $h$ denote the length of the altitudes of $S_{\mathrm{reg}}$, and for
$i=1,2,\ldots,d+1$, let $h_{i}$ denote the distance of $x$ from the facet of
$S_{\mathrm{reg}}$ opposite of $q_{i}$. It is well known that for any $x\in
S_{\mathrm{reg}}$, $\sum_{i=1}^{d+1}h_{i}=h$. Thus, for any $x\in
S_{\mathrm{reg}}$, we have

$\sum_{i=1}^{d+1}|x-q_{i}|\geq\sum_{i=1}^{d+1}(h-h_{i})=dh,$

with equality if and only if $x$ lies on all altitudes of $S_{\mathrm{reg}}$,
or in other words, if $x=o$.

We have shown that
$\operatorname{cr}(\bar{Z})=\operatorname{cr}(Z_{\mathrm{rd}}^{\mathrm{reg}})$
and $w(\bar{Z})>w(Z_{\mathrm{reg}})$. Thus, if we set
$t=\frac{w(Z_{\mathrm{rd}}^{\mathrm{reg}})}{w(\bar{Z})}$, then for
$Z^{\prime}=tZ$ we have $w(Z^{\prime})=w(Z_{\mathrm{rd}}^{\mathrm{reg}})$ and
$\operatorname{cr}(Z^{\prime})<\operatorname{cr}(Z_{\mathrm{rd}}^{\mathrm{reg}})$,
which proves the last statement of Theorem 5. $\Box$

###  3.3. The second intrinsic volume of a zonotope with a given mean width

We start with a simple observation for parallelotopes.

######  Proposition 3.2.

For any $Z\in\mathcal{Z}_{\mathrm{p}}$ with
$w(Z)=w(Z_{\mathrm{p}}^{\mathrm{reg}})$, we have $V_{2}(Z)\leq
V_{2}(Z_{\mathrm{p}}^{\mathrm{reg}})$, with equality if and only if $Z$ is a
cube.

###### Proof.

Let $Z=\sum_{i=1}^{d}[o,p_{i}]$, and set $\lambda_{i}=|p_{i}|$ where, without
loss of generality, we may assume that all the $\lambda_{i}$ are positive..
Then, for any $i\neq j$, we have $|p_{i}\wedge
p_{j}|\leq\lambda_{i}\lambda_{j}$, with equality if and only $p_{i}$ and
$p_{j}$ are orthogonal. Thus, we need to show that if
$\sum_{i=1}^{d}\lambda_{i}$ is fixed, then $A=\sum_{1\leq i<j\leq
d}\lambda_{i}\lambda_{j}$ is maximal if and only if all $\lambda_{i}$ are
equal. But this follows from the Maclaurin inequality (see Lemma 1). $\Box$

In the remaining part of Subsection 3.3, we find the equilateral rhombic
dodecahedra with a given mean width and maximal second intrinsic volume. Our
proof is a modification of a proof of [2, Theorem 3]. We remark that the
‘dual’ problem, that is, minimizing the $i$th intrinsic volume of a zonotope
with a fixed mean width was considered in [29] for the special class of
zonotopes with an isotropic generating measure.

######  Theorem 6.

Let $Z_{\mathrm{rd}}^{\mathrm{reg}}=\sum_{i=1}^{d+1}[o,q_{i}]$, where
$q_{i}\in\mathbb{S}^{d-1}$ for all values of $i$. Then, if
$Z=\sum_{i=1}^{d+1}[o,p_{i}]$ is a rhombic dodecahedron with
$p_{i}\in\mathbb{S}^{d-1}$ for all values of $i$, then

$V_{2}(Z)\geq V_{2}(Z_{\mathrm{rd}}^{\mathrm{reg}}),$

with equality if and only if $Z$ is regular.

###### Proof.

It is well known that for any vectors $x_{1},x_{2},\ldots,x_{k}$ in
$\mathbb{R}^{d}$ with $k\leq d$, the square of the $k$-dimensional volume of
the parallelotope spanned by $x_{1},x_{2},\ldots,x_{k}$ is equal to the
determinant of the Gram matrix of the vectors (see e.g. [22]). In particular,
for any $x_{1},x_{2}\in\mathbb{R}^{d}$, the square of the area of the
parallelogram spanned by $x_{1},x_{2}$ is

$|x_{1}\wedge x_{2}|^{2}=\det([x_{1},x_{2}]^{T}[x_{1},x_{2}])=\langle
x_{1},x_{1}\rangle\langle x_{2},x_{2}\rangle-\langle x_{1},x_{2}\rangle^{2}.$

This and Corollary 1 imply that

$V_{2}(Z)=\sum_{1\leq i<j\leq d+1}\sqrt{\langle p_{i},p_{i}\rangle\langle
p_{j},p_{j}\rangle-\langle p_{i},p_{j}\rangle^{2}}.$

We note that this yields, in particular, that
$V_{2}(Z_{\mathrm{reg}})=\binom{d+1}{2}\cdot\sqrt{\frac{d^{2}-1}{d^{2}}}=\frac{1}{2}(d+1)\sqrt{d^{2}-1}$.
By applying the fact that $Z$ is of unit edge length and also the inequality
between the arithmetic and the quadratic means we obtain that

(12)
$\left(V_{2}(Z)\right)^{2}\leq\binom{d+1}{2}\left(\binom{d+1}{2}-\sum_{1\leq
i<j\leq d+1}\langle p_{i},p_{j}\rangle^{2}\right),$

with equality if and only if $Z$ is regular. In the following part we
determine the minimum value of $\sum_{1\leq i<j\leq d+1}\langle
p_{i},p_{j}\rangle^{2}$.

Since the vectors $p_{1},p_{2},\ldots,p_{d+1}$ are linearly dependent, there
are constants $\lambda_{i}\in\mathbb{R}$, where $i=1,2,\ldots,d+1$, such that
$\sum_{i=1}^{d+1}\lambda_{i}p_{i}=o$ and $\sum_{i=1}^{d+1}\lambda_{i}^{2}>0$.
Without loss of generality, we may assume that
$\sum_{i=1}^{d+1}\lambda_{i}^{2}=1$. Multiplying both sides of this vector
equation by itself and using the equalities $\langle p_{i},p_{i}\rangle=1$, we
obtain that

$0=\sum_{i=1}^{d+1}\lambda_{i}^{2}+2\sum_{1\leq i<j\leq
d+1}\lambda_{i}\lambda_{j}\langle
p_{i},p_{j}\rangle{\color[rgb]{0,0,0}=1+2\sum_{1\leq i<j\leq
d+1}\lambda_{i}\lambda_{j}\langle p_{i},p_{j}\rangle}.$

Thus, by the Cauchy-Schwarz Inequality, we have

${\color[rgb]{0,0,0}1}=-2\sum_{1\leq i<j\leq d+1}\lambda_{i}\lambda_{j}\langle
p_{i},p_{j}\rangle\leq 2\sqrt{\sum_{1\leq i<j\leq
d+1}\lambda_{i}^{2}\lambda_{j}^{2}}\sqrt{\sum_{1\leq i<j\leq d+1}\langle
p_{i},p_{j}\rangle^{2}},$

implying that

(13) $\sum_{1\leq i<j\leq d+1}\langle
p_{i},p_{j}\rangle^{2}\geq\frac{{\color[rgb]{0,0,0}1}}{4\sum_{1\leq i<j\leq
d+1}\lambda_{i}^{2}\lambda_{j}^{2}},$

which yields by Lemma 1 that

$\sum_{1\leq i<j\leq d+1}\langle p_{i},p_{j}\rangle^{2}\geq\frac{2d}{d+1}.$

Combining this inequality with (12), we obtain that

$V_{2}(Z)\leq\sqrt{\binom{d+1}{2}\left(\binom{d+1}{2}-\frac{d+1}{2d}\right)}=\frac{1}{2}(d+1)\sqrt{d^{2}-1}=V_{2}(Z_{\mathrm{reg}}),$

with equality if and only if $Z$ is regular. $\Box$

###  3.4. The squared $k$-volumes of a zonotope

Recall that for any simplex $S\subset\mathbb{R}^{d}$, integer $1\leq k\leq
d-1$, and real number $m\in\mathbb{R}$, $g_{k}^{m}(S)$ denotes the sum of the
$m$th powers of the volumes of the $k$-faces of $S$ [46]. Similarly, Filliman
[19] investigated the problem of finding the zonotopes of maximal volume with
a fixed value of the squared lengths of its generating vectors. These results,
Corollary 1, and the results of Brazitikos and McIntyre in [15] is the
motivation behind our next definition.

######  Definition 1.

Let $Z=\sum_{i=1}^{n}[o,p_{i}]$ be a zonotope in $\mathbb{R}^{d}$ and let
$\alpha>0$. We call the quantity

$V_{k,\alpha}(Z)=\sum_{1\leq i_{1}<i_{2}<\ldots<i_{k}\leq
n}V_{k}^{\alpha}(P(i_{1},\ldots,i_{k}))$

the _total $k$-volume of power $\alpha$_ of $Z$.

Our next result is an Alexandrov-Fenchel type inequality for the total squared
$k$-volumes of rhombic dodecahedra. We remark that an analogous statement for
parallelotopes can be found in [15].

######  Theorem 7.

Let $Z$ be a rhombic dodecahedron in $\mathbb{R}^{d}$. Then, for any $1\leq
k<m\leq d$, the quantity

$\frac{\left(V_{k,2}(Z)\right)^{m}}{\left(V_{m,2}(Z)\right)^{k}}$

is minimal if and only if $Z$ is regular.

First, we prove a slight generalization of Lemma 1.

######  Lemma 4.

Let $1\leq k<m$ be integers, and $x_{1},\ldots,x_{m}\geq 0$ be nonnegative
real numbers. Then

(14)
$\left(\frac{\sigma_{m}^{k}(x_{1},x_{2},\ldots,x_{m})}{\binom{m}{k}}\right)^{\frac{1}{k}}\geq\left(\frac{\sigma_{m}^{\textbf{k+1}}(x_{1},x_{2},\ldots,x_{m})}{\binom{m}{k+1}}\right)^{\frac{1}{k+1}},$

with equality if and only if all $x_{i}$ are equal, or at least $m-k+1$ of
them is zero.

###### Proof.

Since both sides in (14) are continuous functions of their variables, the
inequality in (14) clearly holds. Similarly, the equality case in Lemma 4
holds if all $x_{i}$ are positive. Note that for any values of
$m,k,x_{1},x_{2},\ldots,x_{m}$,
$\sigma_{m}^{k}(x_{1},\ldots,x_{m})=\sigma_{m+1}^{k}(x_{1},\ldots,x_{m},0)$.
Thus, to prove Lemma 4 for all nonnegative $x_{i}$, it is sufficient to show
that if
$\sqrt[k]{\frac{A}{\binom{m}{k}}}\geq\sqrt[k+1]{\frac{B}{\binom{m}{k+1}}}$ for
some reals $A,B>0$ and integers $1\leq k<m$, then
$\sqrt[k]{\frac{A}{\binom{m+1}{k}}}>\sqrt[k+1]{\frac{B}{\binom{m+1}{k+1}}}$.
In other words, we need to show that

(15)
$\left(\frac{\binom{m}{k}}{\binom{m+1}{k}}\right)^{\frac{1}{k}}>\left(\frac{\binom{m}{k+1}}{\binom{m+1}{k+1}}\right)^{\frac{1}{k+1}}$

for all $1\leq k<m$. An elementary computation shows that the inequality in
(15) is equivalent to

$\left(1-\frac{k}{m+1}\right)^{\frac{1}{k}}>\left(1-\frac{k+1}{m+1}\right)^{\frac{1}{k+1}},$

and thus, the assertion follows from the fact that for any $0<a<1$, the
function $x\mapsto\left(1-ax\right)^{\frac{1}{x}}$ is strictly decreasing on
the interval $\left(0,\frac{1}{a}\right]$. $\Box$

###### Proof of Theorem 7.

Consider the zonotope $Z=\sum_{i=1}^{d+1}[o,p_{i}]$. Let $G$ be the Gram
matrix of the vectors $p_{i}$; i.e. $G=[\langle p_{i},p_{j}\rangle]$. Then $G$
is a $(d+1)\times(d+1)$ symmetric, positive semidefinite matrix. Let
$p(\lambda)=\det(G-\lambda E)$ denote the characteristic polynomial of $G$.
Clearly, the coefficient of $\lambda^{d+1-k}$ is equal to the signed sum of
the principal minors of size $k\times k$ [38], where a principal minor of size
$k\times k$ of $G$ is the determinant of a $k\times k$ submatrix of $G$
symmetric to the main diagonal of $G$. On the other hand, it is also well
known that this coefficient is equal to the $k$th elementary symmetric
function of the eigenvalues of $G$. Note that the rank of $G$ is equal to the
dimension of the linear hull of the vectors $v_{i}$, and thus, $0$ is an
eigenvalue of $G$ with multiplicity at least one. Thus, the inequality in
Theorem 7, together with the equality part, follows from Lemma 4 applied for
$d$ eigenvalues of $G$, containing all nonzero ones. $\Box$

We also note that a straightforward modification of the proof of Theorem 7
verifies Conjecture 1 for $p=2$ in the following form.

######  Theorem 8.

Let $n\geq d$ and $Z=\sum_{i=1}^{n}[o,x_{i}]$ be a zonotope in
$\mathbb{R}^{d}$. Then, for any $1\leq k<d$, the quantity

$\left(\frac{V_{k,2}(Z)}{\binom{n}{k}}\right)^{\frac{1}{k}}\geq\left(\frac{V_{k+1,2}(Z)}{\binom{n}{k+1}}\right)^{\frac{1}{k+1}},$

with equality if and only if $n=d$ and $Z$ is a cube, or if the dimension of
$Z$ is at most $k-1$.

##  4\. Isoperimetric problems for zonotopes generated by a large number of
segments

Recall that for any $d\geq 2$ and $n\geq d$, $\mathcal{Z}_{d,n}$ denotes the
family of $d$-dimensional zonotopes generated by $n$ segments. In this section
we investigate isoperimetric problems for zonotopes generated by sufficiently
many segments. It is worth noting that as the Euclidean ball can be approached
arbitrarily well by zonotopes, and in Euclidean space a ball is the solution
of most isoperimetric problems among all convex bodies, the problems discussed
in this section are closely related to the problem of how well a Euclidean
ball can be approached with a zonotope generated by a given number of
segments. The latter problem was extensively studied in the literature,
motivated by the problem of determining the minimum number of directions we
need to estimate the surface area of a convex body up to error $\varepsilon$
from the areas of the projections in the chosen directions. The order of
magnitude of the number of directions was established in most dimensions in a
series of papers [8, 33, 11, 13, 34] by Linhart, Bourgain, Lindestrauss,
Milman and Matoušek. Namely, it is known that for $d\geq 2$ there are
constants $c_{1}=c_{1}(d)$ and $c_{2}=c_{2}(d)$ depending only on $d$ with the
following property: if for any $\varepsilon>0$, $N(\varepsilon)$ denotes the
minimum number $n$ such that there is a zonotope $Z\in\mathcal{Z}_{d,n}$ with
$\mathbf{B}^{d}\subseteq Z\subseteq(1+\varepsilon)\mathbf{B}^{d}$, then

(16) $c_{1}\varepsilon^{\frac{-2(d-1)}{d+2}}\leq
N(\varepsilon)\leq\left\\{\begin{array}[]{l}c_{2}\varepsilon^{\frac{-2(d-1)}{d+2}},\hbox{
if }d=2\hbox{ or }d\geq 5,\\\
c_{2}\left(\varepsilon^{-2}\log|\varepsilon|\right)^{\frac{(d-1)}{d+2}},\hbox{
otherwise}.\end{array}\right.$

Furthermore, it was proved in [12] that the weaker bound on the right-hand
side of (16) can be attained with an equilateral zonotope, i.e. there is a
zonotope $Z$ generated by $n\leq
c_{2}\left(\varepsilon^{-2}\log|\varepsilon|\right)^{\frac{(d-1)}{d+2}}$
segments of equal length such that $\mathbf{B}^{d}\subseteq
Z\subseteq(1+\varepsilon)\mathbf{B}^{d}$.

Similarly like in the previous problem, in our investigation we regard the
dimension $d$ as a fixed parameter, and the number $n$ of generating segments
as a variable. Furthermore, we set

$U_{d}(n)=\left\\{\begin{array}[]{l}\frac{\sqrt{\log
n}}{n^{\frac{d+2}{2d-2}}},\hbox{ if }d=3\hbox{ or }d=4,\\\
\frac{1}{n^{\frac{d+2}{2d-2}}},\hbox{ if }d=2\hbox{ or }d\geq
5.\end{array}\right.$

Our first result is an immediate consequence of (16), and we omit its proof.

######  Theorem 9.

Let $d\geq 2$ be fixed. Then there are positive constants $c=c(d)$ and
$C=C(d)$ depending only on the dimension such that for any $n\geq d+1$,

(17)
$\frac{c}{n^{\frac{d+2}{2d-2}}}\leq\min\left\\{\frac{\operatorname{cr}(Z)}{\operatorname{ir}(Z)}-1:Z\in\mathcal{Z}_{d,n}\right\\}\leq
CU_{d}(n).$

In the remaining part of this section we investigate similar problems. More
specifically, in Subsections 4.1 and 4.2 we estimate the intrinsic volumes of
a zonotope $Z\in\mathcal{Z}_{d,n}$ with a given inradius or circumradius,
respectively. Finally, in Subsection 4.3 we compare two intrinsic volumes of a
zonotope in $\mathcal{Z}_{d,n}$.

###  4.1. Intrinsic volumes of a zonotope with a given inradius

Our main result in this subsection is Theorem 10.

######  Theorem 10.

Let $1\leq i\leq d$. Then there is a positive constant $C=C(d)$ depending only
on $d$ such that for any sufficiently large value of $n$, we have

$\frac{4i}{5dn^{2}}\leq\min\left\\{\frac{V_{i}(Z)}{V_{i}(\mathbf{B}^{d})}-1:Z\in\mathcal{Z}_{d,n},\operatorname{ir}(Z)=1\right\\}\leq
CU_{d}(n).$

To prove it we need the following lemma.

######  Lemma 5.

Let $2\leq d\leq n$. Then we have

$\min\left\\{V_{d}(Z):Z\in\mathcal{Z}_{d,n},\operatorname{ir}(Z)\geq
1\right\\}\geq\kappa_{d}\left(1+\frac{\pi^{2}}{12n^{2}}\right).$

###### Proof.

First, let $d=2$. Consider any $Z\in\mathcal{Z}_{2,n}$ with
$\operatorname{ir}(Z)\geq 1$. Without loss of generality, assume that
$\mathbf{B}^{2}\subset Z$. Then, we clearly have that the minimum of
$V_{2}(Z)$ is the area of the regular $(2n)$-gon circumscribed about
$\mathbf{B}^{2}$. Thus, an elementary computation shows that $V_{2}(Z)\geq
2n\tan\frac{\pi}{2n}=\pi\cdot\frac{\tan\frac{\pi}{2n}}{\frac{\pi}{2n}}$. On
the other hand, from the third order Taylor polynomial of the tangent function
with the Lagrange remainder form, and using the fact that $(\tan x)^{(4)}>0$
for $x\in\left(0,\frac{\pi}{2}\right)$, it follows that $\tan
x>x+\frac{1}{3}x^{3}$. Hence, we obtain that

$V_{2}(Z)>\kappa_{2}\left(1+\frac{\pi^{2}}{12n^{2}}\right).$

In the remaining part of the proof we apply an induction on $d$. Let
$Z\in\mathcal{Z}_{d,n}$ with $\operatorname{ir}(Z)\geq 1$. Assume that
$\mathbf{B}^{d}\subset Z$. For any unit vector $u\in\mathbb{S}^{d-1}$, let
$Z|u^{\perp}$ denote the orthogonal projection of $Z$ onto the hyperplane
through $o$ and with normal vector $u$. Note that for any
$u\in\mathbb{S}^{d-1}$, $Z|u^{\perp}$ is a zonotope in $\mathcal{Z}_{d-1,n}$
satisfying $\operatorname{ir}(Z)\geq 1$. Thus, by the induction hypothesis,
$V_{d-1}(Z|u^{\perp})\geq\kappa_{d-1}\left(1+\frac{\pi^{2}}{12n^{2}}\right)$.
Recall that by Cauchy’s surface area formula [21], we have

$\operatorname{surf}(Z)=\frac{1}{\kappa_{d-1}}\int_{\mathbb{S}^{d-1}}V_{d-1}(Z|u^{\perp})\,du,$

where $\operatorname{surf}(Z)$ denotes the surface area of $Z$. Then, applying
the previous estimate and the fact that the surface area of $\mathbb{S}^{d-1}$
is $d\kappa_{d}$, after simplification we obtain that

(18) $\operatorname{surf}(Z)\geq
d\kappa_{d}\left(1+\frac{\pi^{2}}{12n^{2}}\right).$

Now, let $F$ be any facet of $Z$. Then, by the convexity of $Z$ and since
$\mathbf{B}^{d}\subset Z$, the distance of $\operatorname{aff}(F)$ from $o$ is
at least $1$. Thus,
$V_{d}(\operatorname{conv}(F\cup\\{o\\}))\geq\frac{1}{d}V_{d-1}(F)$, implying
that $\operatorname{surf}(Z)\leq dV_{d}(Z)$. This, combined with (18), yields
the assertion. $\Box$

###### Proof of Theorem 10.

Let $Z$ be an arbitrary zonotope in $\mathcal{Z}_{d,n}$ satisfying
$\operatorname{ir}(Z)=1$. By the Alexandrov-Fenchel inequality for intrinsic
volumes, we have that

(19)
$\left(\frac{V_{d}(Z)}{V_{d}(\mathbf{B}^{d})}\right)^{\frac{1}{d}}\leq\left(\frac{V_{d-1}(Z)}{V_{d-1}(\mathbf{B}^{d})}\right)^{\frac{1}{d-1}}\leq\ldots\leq\frac{V_{1}(Z)}{V_{1}(\mathbf{B}^{d})}.$

Thus,

$\left(\frac{V_{d}(Z)}{V_{d}(\mathbf{B}^{d})}\right)^{\frac{i}{d}}-1\leq\frac{V_{i}(Z)}{V_{i}(\mathbf{B}^{d})}-1\leq\left(\frac{V_{1}(Z)}{V_{1}(\mathbf{B}^{d})}\right)^{i}-1.$

By Lemma 5 and using that $\lim_{x\to 0}\frac{(1+x)^{\alpha}-1}{x}=\alpha$ and
that $\frac{\pi^{2}}{12}>\frac{4}{5}$, for sufficiently large values of $n$,
we have

${\color[rgb]{0,0,0}\frac{4i}{5dn^{2}}\leq\left(\frac{V_{d}(Z)}{V_{d}(\mathbf{B}^{d})}\right)^{\frac{i}{d}}-1\leq\frac{V_{i}(Z)}{V_{i}(\mathbf{B}^{d})}-1.}$

This yields the left-hand side inequality in Theorem 10. To prove the right-
hand side inequality, observe that if $Z$ is a zonotope in $\mathcal{Z}_{d,n}$
with $\mathbf{B}^{d}\subseteq Z\subseteq(1+\varepsilon)\mathbf{B}^{d}$, then
$2\leq w(Z)\leq 2(1+\varepsilon)$. This, combined with Theorem 9, yields that
if $n$ is sufficiently large, then

$\left(\frac{V_{1}(Z)}{V_{1}(\mathbf{B}^{d})}\right)^{i}-1\leq CU_{d}(n)$

for some suitably chosen $Z\in\mathcal{Z}_{d,n}$ with
$\operatorname{ir}(Z)=1$. $\Box$

###  4.2. Intrinsic volumes of a zonotope with a given circumradius

We prove Theorem 11.

######  Theorem 11.

Let $1\leq i\leq d$. Then there is a positive constant $C=C(d)$ such that for
any sufficiently large value of $n$,

$\frac{2i}{5n^{2}}\leq\min\left\\{1-\frac{V_{i}(Z)}{V_{i}(\mathbf{B}^{d})}:Z\in\mathcal{Z}_{d,n},\operatorname{cr}(Z)=1\right\\}\leq
CU_{d}(n).$

######  Lemma 6.

Let $d\geq 2$ and $n\geq d$. Then, for any $Z\in\mathcal{Z}_{n,d}$ with
$\operatorname{cr}(Z)\geq 1$, we have

$1-\frac{V_{1}(Z)}{V_{1}(\mathbf{B}^{d})}\geq\frac{\pi^{2}}{24n^{2}}-\frac{\pi^{4}}{1920n^{4}}.$

###### Proof.

First, we prove the inequality for $d=2$. Let $Z\in\mathcal{Z}_{n,2}$ with
$\operatorname{cr}(Z)\geq 1$, and assume that $Z\subset\mathbf{B}^{2}$. By
Dowker’s theorem, we have that the perimeter of $Z$ is at most equal to the
perimeter of the regular $(2n)$-gon inscribed in $\mathbf{B}^{2}$. Thus, we
have $\operatorname{perim}(Z)\leq 4n\sin\frac{\pi}{2n}$. Note that
$V_{1}(Z)=\frac{1}{2}\operatorname{perim}(Z)$ (cf. [42, p.210]). Thus,
applying the Taylor-expansion of the sine function, we have that

$V_{1}(Z)\leq\pi\left(1-\frac{\pi^{2}}{24n^{2}}+\frac{\pi^{4}}{1920n^{4}}\right).$

Since $V_{1}(\mathbf{B}^{d})=\frac{d\kappa_{d}}{\kappa_{d-1}}$, and in
particular $V_{1}(\mathbf{B}^{2})=\pi$, this implies the assertion for $d=2$.
To prove it for $d>2$, we recall Kubota’s integral recursion formula from [21,
A.46]. To state it, we introduce the notation $\mathcal{G}(d,k)$ for the
Grassmannian manifold of the $k$-dimensional linear subspaces of
$\mathbb{R}^{d}$, $K|S$ for the orthogonal projection of a compact, convex set
$K\subset\mathbb{R}^{d}$ to some $S\in\mathcal{d,k}$, and $\mu_{d,k}(\cdot)$
for the unique Haar probability measure of $\mathcal{G}(d,k)$. Then this
formula states that if $K$ is any compact, convex set in $\mathbb{R}^{d}$, and
$1\leq i\leq k\leq d-1$, then

(20)
$V_{i}(K)=\frac{\binom{d}{i}\kappa_{k-i}\kappa_{d}}{\binom{k}{i}\kappa_{d-i}\kappa_{k}}\int_{\mathcal{G}_{d,k}}V_{i}(K|S)\,d\mu_{d,k}(S).$

Let $Z\in\mathcal{Z}_{d,n}$ be a zonotope with $\operatorname{cr}(Z)\leq 1$.
Then, for any $S\in\mathcal{G}_{d,2}$, $Z|S$ is a zonotope in $Z_{2,n}$ with
$\operatorname{cr}(Z|S)\leq 1$. Thus,
$V_{1}(Z|S)\leq\pi\left(1-\frac{\pi^{2}}{24n^{2}}+\frac{\pi^{4}}{1920n^{4}}\right)$
for any $S\in\mathcal{G}_{d,2}$. Now, applying (20) for $Z$ with $i=1$ and
$k=2$, we obtain that

$V_{1}(Z)\geq\frac{d\kappa_{d}}{\kappa_{d-1}}\left(1-\frac{\pi^{2}}{24n^{2}}+\frac{\pi^{4}}{1920n^{4}}\right),$

from which the assertion readily follows. $\Box$

###### Proof of Theorem 11.

By the Alexandrov-Fenchel inequality for any zonotope $Z\in\mathcal{Z}_{d,n}$
with $\operatorname{cr}(Z)=1$, we have

$1-\left(\frac{V_{d}(Z)}{V_{d}(\mathbf{B}^{d})}\right)^{\frac{i}{d}}\geq
1-\frac{V_{i}(Z)}{V_{i}(\mathbf{B}^{d})}\geq
1-\left(\frac{V_{1}(Z)}{V_{1}(\mathbf{B}^{d})}\right)^{i}.$

Thus, the inequalities in Theorem 11 follow by an argument analogous to that
in the proof of Theorem 10. $\Box$

######  Remark 5.

It was shown in [3, Theorem 1] (and remarked also in [2]) that

$M_{n}^{1}(\mathbb{S}^{d-1})=n\mu_{d,1}+o\left(\frac{n}{\sqrt{d}}\right)$

if $n,d\to\infty$ and $n=\omega\left(d^{2}\log d\right)$, where
$\mu_{d,1}=\frac{\Gamma\left(\frac{d}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)}$.
Theorem 11 yields another estimate for this quantity. Namely, for any fixed
$d\geq 2$, there is a constant $C=C(d)$ such that

$\left|M_{n}^{1}(\mathbb{S}^{d-1})-n\mu_{d,1}\right|\leq
Cn^{\frac{d-4}{2d-2}}\sqrt{\log n}.$

###### Proof.

By Theorem 11, there is a zonotope $Z\in\mathcal{Z}_{d,n}$ with
$\operatorname{cr}(Z)=1$ satisfying

$1-\frac{V_{1}(Z)}{V_{1}(\mathbf{B}^{d})}\leq C\frac{\sqrt{\log
n}}{n^{\frac{d+2}{2d-2}}}$

where $C>0$ may depend on $d$. Here, by [12], we may assume that $Z$ is an
equilateral zonotope, i.e. that all generating segments of $Z$ are of length
$t$ for some $t>0$. The definition of intrinsic volume and an elementary
computation shows that
$V_{1}(\mathbf{B}^{d})=\frac{d\kappa_{d}}{\kappa_{d-1}}=\frac{2\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)}{\Gamma\left(\frac{d}{2}\right)}=\frac{2}{\mu_{d,1}}$.
Let $Z^{\prime}=\sum_{i=1}^{n}[o,p_{i}]=\frac{1}{t}Z$. Then
$V_{1}(Z^{\prime})=n$, and
$\operatorname{cr}(Z^{\prime})=\frac{1}{t}=\frac{1}{2}\max_{v\in\mathbb{S}^{d-1}}\sum_{i=1}^{n}|\langle
v,p_{i}\rangle|$, which implies that

$1-\frac{n\mu_{d,1}}{2\operatorname{cr}(Z^{\prime})}\leq C\frac{\sqrt{\log
n}}{n^{\frac{d+2}{2d-2}}}.$

From this inequality and the inequality $\frac{1}{1-x}<1+2x$ holding for any
sufficiently small $x>0$, we obtain that

$2\operatorname{cr}(Z^{\prime})\leq n\mu_{d,1}\left(1+2C\frac{\sqrt{\log
n}}{n^{\frac{d+2}{2d-2}}}\right),$

which yields the assertion. $\Box$

###  4.3. Comparing two intrinsic volumes of a zonotope

Our main result here is Theorem 12.

######  Theorem 12.

Let $1\leq i<k\leq d$. Then there are positive constants $c,C$ depending only
on $d$ such that for any sufficiently large value of $n$,

(21)
$\frac{c}{n^{\frac{(d+2)(d+3)}{4d-4}}}\leq\min\left\\{\frac{\left(V_{i}(Z)\right)^{\frac{1}{i}}}{\left(V_{k}(Z)\right)^{\frac{1}{k}}}-\frac{\left(V_{i}(\mathbf{B}^{d})\right)^{\frac{1}{i}}}{\left(V_{k}(\mathbf{B}^{d})\right)^{\frac{1}{k}}}:Z\in\mathcal{Z}_{d,n}\right\\}\leq\frac{C}{n}.$

Furthermore, there is a constant $\bar{c}>0$ depending on $d$ such that

(22)
$\frac{\bar{c}}{n^{2}}\leq\min\left\\{\frac{\left(V_{d-1}(Z)\right)^{\frac{1}{{d-1}}}}{\left(V_{d}(Z)\right)^{\frac{1}{d}}}-\frac{\left(V_{d-1}(\mathbf{B}^{d})\right)^{\frac{1}{d-1}}}{\left(V_{d}(\mathbf{B}^{d})\right)^{\frac{1}{d}}}:Z\in\mathcal{Z}_{d,n}\right\\}.$

To prove this theorem, we need some preparation. In the following, let
$\omega_{d}$ denote the surface area of the Euclidean sphere
$\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$.

######  Remark 6.

For any $d\geq 2$, we have

$\frac{V_{d}(\mathbf{B}^{d})}{\left(V_{1}(\mathbf{B}^{d})\right)^{d}}=\frac{2\omega_{d+1}^{d-1}}{\omega_{d}^{d}d!}.$

###### Proof.

Note that $\omega_{d}=d\kappa_{d}$, and
$\kappa_{d}=\frac{\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}+1\right)}$,
implying also the identity $\kappa_{d+1}=\frac{2\pi}{d+1}\kappa_{d-1}$.
Furthermore, it is well known that
$V_{1}(\mathbf{B}^{d})=\frac{d\kappa_{d}}{\kappa_{d-1}}$. Applying these
identities, we have

$\frac{2\omega_{d+1}^{d-1}}{\omega_{d}^{d}d!}\cdot\frac{\left(V_{1}(\mathbf{B}^{d})\right)^{d}}{V_{d}(\mathbf{B}^{d})}=\frac{2(d+1)^{d-1}\kappa_{d+1}^{d-1}}{d!\kappa_{d-1}^{d}\kappa_{d}}=\frac{2^{d}\pi^{d-1}}{d!\kappa_{d}\kappa_{d-1}}=\frac{2^{d}\Gamma\left(\frac{d+2}{2}\right)\Gamma\left(\frac{d+1}{2}\right)}{d!\sqrt{\pi}}=1,$

where in the last step we used the Legendre duplication formula
$\Gamma(z)\Gamma\left(z+\frac{1}{2}\right)=2^{1-2z}\sqrt{\pi}\Gamma(2z)$ for
$z\in\mathbb{C}$, and the identity $\Gamma(k)=(k-1)!$ for all positive
integers $k$. $\Box$

Our next lemma is a special case of [43, Theorem 8.2.3], and hence, we omit
its proof.

######  Lemma 7.

Let $p_{1},\ldots,p_{d}\in\mathbb{S}^{d-1}$ be unit vectors chosen
independently according to the uniform probability distribution on
$\mathbb{S}^{d-1}$. Then the expected value of $|p_{1}\wedge
p_{2}\wedge\ldots\wedge p_{d}|$ is
$\frac{2\omega_{d+1}^{d-1}}{\omega_{d}^{d}}$.

###### Proof of Theorem 12.

First, we prove the lower bound in (21) for $i=1$ and $k=2$. Let
$Z\in\mathcal{Z}_{d,n}$. Since the statement is invariant under rescaling $Z$,
we can assume that $V_{1}(Z)=V_{1}(\mathbf{B}^{d})$ and that $Z$ is centered
at the origin. Then the Steiner ball of $Z$, defined as the ball centered at
the Steiner point of $Z$ and having mean width equal to that of $Z$ [42],
coincides with $\mathbf{B}^{d}$. Our proof in this case is based on a
stability version of the Alexandrov-Fenchel inequality, stated in (7.123) of
[42] as a consequence of [42, Theorem 7.6.6] and implying that
$\frac{1}{\kappa_{d}}\left(W_{d-1}(Z)\right)^{2}-W_{d-2}(Z)\geq\frac{d+1}{d(d-1)}\left(\delta_{2}(Z,\mathbf{B}^{d})\right)^{2}$,
where $W_{i}(K)$ denotes the $i$th quermassintegral of the convex body $K$,
and
$\delta_{2}(K,L)=\sqrt{\int_{u\in\mathbb{S}^{d-1}}|h_{K}(u)-h_{L}(u)|^{2}\,du}$,
with $h_{K}$ and $h_{L}$ denoting the support functions of the convex bodies
of $K$ and $L$, respectively. Applying the well known relation
$W_{d-i}(Z)=\frac{\kappa_{d-i}}{\binom{d}{i}}V_{i}(Z)$ for $i=1,2$, and using
$W_{i}(\mathbf{B}^{d})=\kappa_{d}$ for all values of $i$, we obtain that

$\frac{\left(V_{1}(Z)\right)^{2}}{\left(V_{1}(\mathbf{B}^{d})\right)^{2}}-\frac{V_{2}(Z)}{V_{2}(\mathbf{B}^{d})}=1-\frac{V_{2}(Z)}{V_{2}(\mathbf{B}^{d})}\geq\frac{d+1}{\kappa_{d}d(d-1)}\left(\delta_{2}(K,\mathbf{B}^{d})\right)^{2}.$

We note that by Lemma 7.6.5 of [42] (see also Lemmas 1 and 2 in [23]), we have

$\left(\delta_{2}(Z,\mathbf{B}^{d})\right)^{2}\geq
c^{\prime}\left(\delta(Z,\mathbf{B}^{d})\right)^{\frac{d+3}{2}},$

where $\delta(\cdot,\cdot)$ denotes the Hausdorff distance, and $c^{\prime}>0$
is a constant depending only on the dimension $d$. If for some
$\varepsilon>0$, $\delta(Z,\mathbf{B}^{d})\geq\varepsilon$ for all
$Z\in\mathcal{Z}_{d,n}$ with Steiner ball $\mathbf{B}^{d}$ and for all
sufficiently large values of $n$, we are done. Thus, we may assume that
$\delta(Z,\mathbf{B}^{d})\geq\frac{1}{2}$, implying that
$\frac{1}{2}\leq\operatorname{ir}(Z)$. Observe that since the Steiner ball of
$Z$ is $\mathbf{B}^{d}$, we have
$\delta(Z,\mathbf{B}^{d})\geq\frac{1}{2}\left(\operatorname{cr}(Z)-\operatorname{ir}(Z)\right)\geq\frac{1}{4}\left(\frac{\operatorname{cr}(Z)}{\operatorname{ir}(Z)}-1\right)$.
Hence, from Theorem 9 it follows that

$1-\frac{V_{2}(Z)}{V_{2}(\mathbf{B}^{d})}\geq
c^{\prime\prime}\left(\delta(Z,\mathbf{B}^{d})\right)^{\frac{d+3}{2}}\geq\frac{c^{\prime\prime\prime}}{n^{\frac{(d+2)(d+3)}{4d-4}}}$

for some constant $c^{\prime\prime\prime}>0$ depending on the dimension. After
some elementary calculations, this implies the required bound.

Now we prove the lower bound in (21) for any value of $i$ with $k=i+1$, and
observe that this clearly yields the lower bound in (21) for all values of $i$
and $k$. By the Alexandrov-Fenchel inequality (see e.g. [7, Theorem 10 (ii)],
for any $2\leq j\leq d-1$ and any $Z\in\mathcal{Z}_{d,n}$, we have
$\left(\frac{V_{j}(Z)}{V_{j}(\mathbf{B}^{d})}\right)^{2}\geq\frac{V_{j-1}(Z)}{V_{j-1}(\mathbf{B}^{d})}\frac{V_{j+1}(Z)}{V_{j+1}(\mathbf{B}^{d})}$,
implying
$\frac{\left(V_{j}(Z)/V_{j}(\mathbf{B}^{d})\right)^{1/j}}{\left(V_{j+1}(Z)/V_{j+1}(\mathbf{B}^{d})\right)^{1/(j+1)}}\geq\left(\frac{\left(V_{j-1}(Z)/V_{j-1}(\mathbf{B}^{d})\right)^{1/(j-1)}}{\left(V_{j}(Z)/V_{j}(\mathbf{B}^{d})\right)^{1/j}}\right)^{\frac{j-1}{j+1}}$.
From this, it follows that

$\frac{\left(V_{i}(Z)/V_{i}(\mathbf{B}^{d})\right)^{1/i}}{\left(V_{i+1}(Z)/V_{i+1}(\mathbf{B}^{d})\right)^{1/(i+1)}}\geq\left(\frac{V_{1}(Z)/V_{1}(\mathbf{B}^{d})}{\sqrt{V_{2}(Z)/V_{2}(\mathbf{B}^{d})}}\right)^{\frac{2}{i(i+1)}}.$

We investigate the left-hand side. Similarly like in the previous paragraph,
we may assume that
$\frac{V_{1}(Z)/V_{1}(\mathbf{B}^{d})}{\sqrt{V_{2}(Z)/V_{2}(\mathbf{B}^{d})}}\leq
2$. Under this assumption, we obtain that

$\left(\frac{V_{1}(Z)/V_{1}(\mathbf{B}^{d})}{\sqrt{V_{2}(Z)/V_{2}(\mathbf{B}^{d})}}\right)^{\frac{2}{i(i+1)}}=\left(1+\frac{V_{1}(Z)/\sqrt{V_{2}(Z)}}{V_{1}(\mathbf{B}^{d})/\sqrt{V_{2}(\mathbf{B}^{d})}}-1\right)^{\frac{2}{i(i+1)}}\geq$
$\geq
1+\frac{\sqrt{V_{2}(\mathbf{B}^{d})}}{V_{1}(\mathbf{B}^{d})}\cdot\frac{\frac{V_{1}(Z)}{\sqrt{V_{2}(Z)}}-\frac{V_{1}(\mathbf{B}^{d})}{\sqrt{V_{2}(\mathbf{B}^{d})}}}{2^{(i^{2}+i)/2}-1}\geq
1+\frac{\hat{c}}{n^{\frac{(d+2)(d+3)}{4d-4}}}$

for some $\hat{c}>0$ depending on $d$. From this a similar computation yields
the lower bound in (21) for $k=i+1$.

Now we prove the inequality in (22). Let $Q$ be the convex polytope with the
same outer unit normals as $Z$ and circumscribed about $\mathbf{B}^{d}$. Then,
by Lindelöf’s theorem [24],
$\frac{\left(V_{d-1}(Z)\right)^{\frac{1}{d-1}}}{\left(V_{d}(Z)\right)^{\frac{1}{d}}}\geq\frac{\left(V_{d-1}(Q)\right)^{\frac{1}{d-1}}}{\left(V_{d}(Q)\right)^{\frac{1}{d}}}$.
On the other hand, applying the volume formula of a cone for the convex hull
of each facet of $Q$ with $\\{o\\}$, we obtain that
$\operatorname{surf}(Q)=dV_{d}(Q)$, implying $V_{d-1}(Q)=\frac{d}{2}V_{d}(Q)$.
Thus, applying an asymptotic estimate on the minimal volume of a convex
polytope with a given number of facets and circumscribed about
$\mathbf{B}^{d}$ [24, 7], we obtain that

(23)
$\frac{\left(V_{d-1}(Z)\right)^{\frac{1}{d-1}}}{\left(V_{d}(Z)\right)^{\frac{1}{d}}}\geq\left(\frac{d}{2}\right)^{\frac{1}{d-1}}\cdot\left(V_{d}(Q)\right)^{\frac{1}{d(d-1)}}\geq\left(\frac{d}{2}\right)^{\frac{1}{d-1}}\kappa_{d}^{\frac{1}{d(d-1)}}\cdot\left(1+\frac{\gamma}{\left(\binom{n}{d-1}\right)^{\frac{2}{d-1}}}\right)^{\frac{1}{d(d-1)}}.$

for any zonotope $Z\in\mathcal{Z}_{d,n}$. Since
$\left(\frac{d}{2}\right)^{\frac{1}{d-1}}\kappa_{d}^{\frac{1}{d(d-1)}}=\frac{\left(V_{d-1}(\mathbf{B}^{d})\right)^{\frac{1}{d-1}}}{\left(V_{d}(\mathbf{B}^{d})\right)^{\frac{1}{d}}}$,
and
$\binom{n}{d-1}^{\frac{2}{d-1}}\leq\frac{n^{2}}{((d-1)!)^{\frac{2}{d-1}}}$,
this implies the bound in (22).

Finally, to prove the upper bound in (21), first, observe that by (19), for
any $1\leq i\leq k\leq d$, and zonotope $Z$, we have

(24)
$\frac{\left(V_{i}(Z)\right)^{\frac{1}{i}}/\left((V_{i}(\mathbf{B}^{d})\right)^{\frac{1}{i}}}{\left(V_{k}(Z)\right)^{\frac{1}{k}}/\left((V_{k}(\mathbf{B}^{d})\right)^{\frac{1}{k}}}\leq\frac{V_{1}(Z)/V_{1}(\mathbf{B}^{d})}{\left(V_{d}(Z)\right)^{\frac{1}{d}}/\left(V_{d}(\mathbf{B}^{d})\right)^{\frac{1}{d}}}.$

We give an upper bound on
$\min\left\\{\frac{V_{1}(Z)}{\left(V_{d}(Z)\right)^{\frac{1}{d}}}:Z\in\mathcal{Z}_{d,n}\right\\}$.
To do it we choose $n$ unit vectors $p_{i}\in\mathbb{S}^{d-1}$ independently
and using uniform distribution, and set $Z=\sum_{i=1}^{n}[o,p_{i}]$. Then, by
the linearity of expectation and Lemma 7, we have that the expected value of
$V_{d}(Z)$ is $\binom{n}{d}\cdot\frac{2\omega_{d+1}^{d-1}}{\omega_{d}^{d}}$.
Since $V_{1}(Z)=n$ for any such zonotope and by Remark 6, there is some
$Z\in\mathcal{Z}_{d,n}$ such that

$\frac{V_{1}(Z)}{\left(V_{d}(Z)\right)^{\frac{1}{d}}}\leq
n\cdot\left(\frac{\omega_{d}^{d}}{2\binom{n}{d}\omega_{d+1}^{d-1}}\right)^{\frac{1}{d}}=\frac{n}{\sqrt[d]{(n(n-1)\ldots(n-d+1))}}\frac{V_{1}(\mathbf{B}^{d})}{\left(V_{d}(\mathbf{B}^{d})\right)^{\frac{1}{d}}}\leq$
$\leq\left(1+\frac{d-1}{n-d+1}\right)\frac{V_{1}(\mathbf{B}^{d})}{\left(V_{d}(\mathbf{B}^{d})\right)^{\frac{1}{d}}}\leq\left(1+\frac{2d}{n}\right)\frac{V_{1}(\mathbf{B}^{d})}{\left(V_{d}(\mathbf{B}^{d})\right)^{\frac{1}{d}}},$

for any $n\geq 2d$. Combining it with (24), we obtain that

$\frac{\left(V_{i}(Z)\right)^{\frac{1}{i}}}{\left(V_{k}(Z)\right)^{\frac{1}{k}}}\leq\frac{\left((V_{i}(\mathbf{B}^{d})\right)^{\frac{1}{i}}}{\left((V_{k}(\mathbf{B}^{d})\right)^{\frac{1}{k}}}\cdot\left(1+\frac{2d}{n}\right),$

from which the upper bound in Theorem 12 readily follows. $\Box$

######  Remark 7.

We remark that if $Z_{0}\in\mathcal{Z}_{d,n}$ satisfies the condition that
$\operatorname{ir}(Z_{0})=1$ and

$V_{i}(Z_{0})=\min\\{V_{i}(Z):Z\in\mathcal{Z}_{d,n},\operatorname{ir}(Z)=1\\},$

then by (19) and Theorem 10, for any $1\leq i<k\leq d$ we have

$0\leq\left(\frac{V_{i}(Z)}{V_{d}(\mathbf{B}^{d})}\right)^{\frac{1}{i}}-\left(\frac{V_{k}(Z)}{V_{k}(\mathbf{B}^{d})}\right)^{\frac{1}{k}}\leq\left(\frac{V_{i}(Z)}{V_{d}(\mathbf{B}^{d})}\right)^{\frac{1}{i}}-1\leq
CU_{d}(n).$

for some constant $C>0$ and sufficiently large value of $n$.

##  5\. Remarks and open problems

In Theorem 3, we have shown that a regular rhombic dodecahedron has minimal
circumradius among the regular dodecahedra of the same volume.

######  Problem 1.

Is it true that the circumradius of a unit volume rhombic dodecahedron $Z$ is
minimal only if $Z$ is regular?

######  Problem 2.

Let $1\leq k<d$. Find the elements $Z\in\mathcal{Z}_{d,d+1}$ with
$\operatorname{cr}(Z)=1$ and maximal $k$th intrinsic volume.

We note that in Problem 2, by Theorem 3, $Z$ is a regular rhombic dodecahedron
if $k=d$, and by Theorem 5 $Z$ is not regular if $k=1$ and $d$ is odd.

######  Problem 3.

Prove or disprove that for any $1<k<d$, the elements $Z\in\mathcal{Z}_{d,d+1}$
with $w(Z)=1$ and maximal value of $V_{k}(Z)$ are regular rhombic dodecahedra.

######  Problem 4.

Find exact orders of magnitudes in the problems discussed in Section 4.

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