# Discrete isoperimetric problems in spaces of constant curvature

Bushra Basit  and  Zsolt Lángi  Bushra Basit, Department of Geometry, Budapest
University of Technology and Economics,
Műegyetem rkp. 3., H-1111 Budapest, Hungary<EMAIL_ADDRESS>Zsolt
Lángi, Department of Geometry, Budapest University of Technology and
Economics, and MTA-BME Morphodynamics Research Group,
Műegyetem rkp. 3., H-1111 Budapest, Hungary<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to prove isoperimetric inequalities for simplices and
polytopes with $d+2$ vertices in Euclidean, spherical and hyperbolic
$d$-space. In particular, we find the minimal volume $d$-dimensional
hyperbolic simplices and spherical tetrahedra of a given inradius.
Furthermore, we investigate the properties of maximal volume spherical and
hyperbolic polytopes with $d+2$ vertices with a given circumradius, and the
hyperbolic polytopes with $d+2$ vertices with a given inradius and having a
minimal volume or minimal total edge length. Finally, for any $1\leq k\leq d$,
we investigate the properties of Euclidean simplices and polytopes with $d+2$
vertices having a fixed inradius and a minimal volume of its $k$-skeleton. The
main tool of our investigation is Euclidean, spherical and hyperbolic Steiner
symmetrization.

###### Key words and phrases:

spherical polytope, hyperbolic polytope, volume, total edge length, discrete
isoperimetric problem, Steiner symmetrization

###### 2010 Mathematics Subject Classification:

52B60, 52A55, 52B11

Partially supported by the BME Water Sciences & Disaster Prevention TKP2020
Institution Excellence Subprogram, grant no. TKP2020 BME-IKA-VIZ and the NKFIH
grant K119670.

##  1\. Introduction

The classical Discrete Isoperimetric Inequality, stating that among convex
$n$-gons of unit area in the Euclidean plane $\mathbb{E}^{2}$, the ones with
minimal perimeter are the regular ones, was already observed by Zenodorus [3].
Since that time, many similar problems, often called isoperimetric problems,
have appeared in the literature [6], asking about the extremal value of a
geometric quantity in a certain family of convex polytopes. In particular,
results about convex polyhedra in Euclidean $3$-space with a given number of
vertices or faces and having a fixed inradius or circumradius can be found,
e.g. in [9] or [1].

Much less is known about convex polytopes with a given number of vertices in
the $d$-dimensional Euclidean space $\mathbb{E}^{d}$. Among the known results
for polytopes with $d+2$ vertices, which in most problems seems to be the
first interesting case, we can mention the paper [5] of Böröczky and Böröczky
Jr. finding the minimum surface area polytopes of unit volume in this family,
[15] of Klein and Wessler determining the maximum volume polytopes with unit
diameter, and [10] of G.Horváth and the second named author about maximum
volume polytopes with unit circumradius. We note that both [15] and [10]
contain partial results for polytopes with $d+3$ vertices (see also [16]).

In the authors’ knowledge, in hyperbolic and spherical spaces there are just a
few examples of solutions of isoperimetric problems. In particular,
Peyerimhoff [18] proved that among hyperbolic simplices inscribed in a given
ball, the regular ones have maximal volume, and Böröczky [4] proved the
analogous statement for spherical simplices.

Our goal in this paper is to examine isoperimetric problems in $d$-dimensional
Euclidean, hyperbolic and spherical space for polytopes with $d+1$ or $d+2$
vertices. Our main tool is the Steiner symmetrization of convex sets, whose
hyperbolic and spherical counterparts were introduced by Schneider [21] and
Böröczky [4], respectively.

In Section 2, we introduce the necessary notation and prove some lemmas needed
in our investigation, describing also the properties of Steiner
symmetrization. In Section 3 we present our main results. In Section 4 we
apply our methods to prove statements for measures generated by rotationally
symmetric density functions. Finally, in Section 5 we present some additional
remarks and pose some open problems.

##  2\. Preliminaries

Throughout the paper we denote the $d$-dimensional Euclidean, hyperbolic and
spherical space by by $\mathbb{E}^{d},\mathbb{H}^{d}$ and $\mathbb{S}^{d}$,
respectively. We regard $\mathbb{S}^{d}$ as the unit sphere centered at the
origin $o$ of the space $\mathbb{E}^{d+1}$.

Let $\mathbb{M}\in\\{\mathbb{E}^{d},\mathbb{H}^{d},\mathbb{S}^{d}\\}$. For any
two points $p,q\in\mathbb{M}$, which are not antipodal if
$\mathbb{M}=\mathbb{S}^{d}$, we denote the shortest geodesic connecting $p$
and $q$ by $[p,q]$, and call it the _closed segment with endpoints $p$ and
$q$_; the distance $d(p,q)$ of $p$ and $q$ is defined as the length of
$[p,q]$. If $\mathbb{M}=\mathbb{E}^{d}$, then for any point $x$ its distance
from the origin $o$ is the Euclidean norm of $x$, which we denote by $||x||$.

We say that $K\subset\mathbb{M}$ is convex if for any two points $p,q\in K$,
we have $[p,q]\subseteq K$; here, if $\mathbb{M}=\mathbb{S}^{d}$, we also
assume that $K$ is contained in an open hemisphere of $\mathbb{S}^{d}$.
Furthermore, for any $X\subseteq\mathbb{M}$, where $X$ is contained in an open
hemisphere if $\mathbb{M}=\mathbb{S}^{d}$, the intersection of all convex sets
containing $X$ is a convex set, called the _convex hull_ of $X$, and denoted
by $\operatorname{conv}(X)$. A compact, convex set with nonempty interior is
called a _convex body_. A convex polytope in $\mathbb{M}$ is the convex hull
of finitely many points. A face of a convex polytope $P$ is the intersection
of the polytope with a supporting hyperplane of $P$, $0$-dimensional faces of
$P$ are called vertices of $P$, and we denote the vertex set of $P$ with
$V(P)$.

Let $P\subset\mathbb{M}$ be a $d$-dimensional convex polytope. The radius of a
largest ball contained in $P$ is called the _inradius_ of $P$, and is denoted
by $\operatorname{ir}(P)$, and the radius of a smallest ball containing $P$ is
called the _circumradius_ of $P$, denoted by $\operatorname{cr}(P)$. For any
$1\leq k\leq d-1$, the _$k$ -skeleton_ of $P$ is the union of its
$k$-dimensional faces; we denote it by $\operatorname{skel}_{k}(P)$. For any
$1\leq k\leq d$, we denote $k$-dimensional volume by
$\operatorname{vol}_{k}(\cdot)$, and call
$\operatorname{vol}_{k}(\operatorname{skel}_{k}(P))$ the _total $k$-content_
of $P$, or if $k=1$, then the _total edge length_ of $P$.

Note that in the projective ball model of $\mathbb{H}^{n}$, hyperbolic line
segments are represented by Euclidean line segments, implying that hyperbolic
convex sets are exactly those represented by Euclidean convex sets.
Furthermore, applying central projection onto a tangent hyperplane of
$\mathbb{S}^{d}$, the same holds in spherical space as well. This shows that
Remarks 1-4, which are well known in $\mathbb{E}^{d}$, hold in
$\mathbb{H}^{d}$ and $\mathbb{S}^{d}$ as well. We note that for
$\mathbb{M}=\mathbb{E}^{d}$, Remarks 1 and 2 can be found in [12, Section 6.1]
(see also [10]).

######  Remark 1.

Every $d$-dimensional simplicial polytope $P\subset\mathbb{M}$ with $d+2$
vertices is the convex hull of two simplices $S_{1},S_{2}$ with dimensions
$\dim S_{1}=k$ and $\dim S_{2}=d-k$ with some $0\leq
k\leq\lfloor\frac{d}{2}\rfloor$, respectively, such that $S_{1}\cap
S_{2}=\\{p\\}$ for some point $p$ in the relative interior of both $S_{1}$ and
$S_{2}$. Furthermore, every $d$-dimensional polytope $Q\subset\mathbb{M}$ is
the convex hull of $r$ points and a $(d-r)$-dimensional simplicial polytope
with $d-r+2$ vertices, for some $0\leq r\leq d-2$.

We note that in Remark 1, $S_{1}$ and $\operatorname{conv}(S_{2}\cup Q)$ are
two simplices which intersect at a point which is a relative interior point of
$S_{1}$ and a point of $\operatorname{conv}(S_{2}\cup Q)$.

######  Remark 2.

The combinatorial structure of a convex polytope with a few vertices can be
described e.g. by a _Gale diagram_ of the polytope. If $P$ is a
$d$-dimensional convex polytope with $d+2$ vertices, and
$P=\operatorname{conv}(S_{1}\cup S_{2}\cup\\{p_{1},\ldots,p_{r}\\})$, where
$\dim S_{1}=k_{1}$, $\dim S_{2}=k_{2}$ with $k_{1}+k_{2}+r=d$ and
$k_{1},k_{2}\geq 1$, then $P$ can be represented by a multiset consisting of
the points $-1,0,1$ on the real line, with multiplicities $k_{1}+1,r,k_{2}+1$.
Here the elements equal to $-1$ represent the vertices of $S_{1}$, those equal
to $1$ represent the vertices of $S_{2}$, and those equal to $0$ represent the
$p_{i}$. In this representation, a subset of the vertex set of $P$ is the
vertex set of a face of $P$ if and only if the complement of the vertex set of
$P$ is associated to points in the diagram whose convex hull contains $0$ in
its relative interior. In particular, if $k_{1},k_{2}\geq 2$, then every pair
of vertices of $P$ is connected by an edge, and the same holds for any pair
apart from the vertices of $S_{1}$ and/or $S_{2}$ if $k_{1}=1$ and/or
$k_{2}=1$, respectively.

######  Remark 3.

Let $P=\operatorname{conv}(S_{1}\cup S_{2})$ be the convex hull of two
simplices with $\dim S_{1}+\dim S_{2}=d$ such that the subspaces generated by
them intersect at a single point $p$ which belongs to $S_{1}\cap S_{2}$, and
is different from any vertex of $S_{1}$ and $S_{2}$. Then, for any vertices
$q_{1},q_{2}$ of $S_{1}$, there is a hyperplane $H$ containing
$V(P)\setminus\\{q_{1},q_{2}\\}$ and intersecting $[q_{1},q_{2}]$.

###### Proof.

Observe that $V(P)\setminus\\{q_{1},q_{2}\\}$ contains $d$ points, and thus,
there is a hyperplane containing it. We show that this hyperplane can be
chosen in such a way that it intersects $[q_{1},q_{2}]$. Indeed, any
hyperplane $H^{\prime}$ containing $V(P)\setminus\\{q_{1},q_{2}\\}$ satisfies
$S_{2}\subset H^{\prime}$, implying also $p\in H^{\prime}$. Furthermore, since
$V(S_{1})\setminus\\{q_{1},q_{2}\\}\subset H^{\prime}\cap L_{1}$, if
$p\notin\operatorname{conv}(V(S_{1})\setminus\\{q_{1},q_{2}\\})$, then there
is a unique hyperplane $H^{\prime}$ containing
$V(P)\setminus\\{q_{1},q_{2}\\}$, and this hyperplane intersects
$[q_{1},q_{2}]$. If
$p\in\operatorname{conv}(V(S_{1})\setminus\\{q_{1},q_{2}\\})$, then the
subspace spanned by $V(P)\setminus\\{q_{1},q_{2}\\}$ is $(d-2)$-dimensional,
and thus, we may choose $H^{\prime}$ such that it contains a point of
$[q_{1},q_{2}]$. ∎

######  Remark 4.

Let $H$ be a hyperplane of $\mathbb{M}$. Let $Q$ be a compact, convex set in
$H$, and let $[p_{1},p_{2}]$ be a segment such that $H\cap[p_{1},p_{2}]$ is a
singleton $\\{p\\}$. Let $P=\operatorname{conv}(Q\cup[p_{1},p_{2}])$. Then,
for any plane $F$ containing $[p_{1},p_{2}]$, $F\cap P$ is either
$[p_{1},p_{2}]$, or a triangle $\operatorname{conv}\\{q,p_{1},p_{2}\\}$, or a
quadrangle $\operatorname{conv}\\{q_{1},q_{2},p_{1},p_{2}\\}$, where
$q,q_{1},q_{2}$ are relative boundary points of $Q$.

###  2.1. Euclidean Steiner symmetrization

######  Definition 1.

Let $H$ be a hyperplane in $\mathbb{E}^{d}$. The _Steiner symmetrization
$\sigma$_ with respect to $H$ is the geometric transformation which assigns to
any convex body $K\subset\mathbb{E}^{d}$ the unique compact set $K^{\prime}$
symmetric to $H$ with the property that for any line $L$ orthogonal to $H$,
$K\cap L$ and $K^{\prime}\cap L$ are both nondegenerate segments of equal
length, or singletons, or the empty set (see Figure 1). In this case we call
the set $\sigma(K)$ the _Steiner symmetral_ of $K$.

Figure 1. Euclidean Steiner symmetrization of a plane convex body $K$ with
respect to a horizontal line $H$. Sections with the same vertical lines are of
equal length.

######  Lemma 1.

Let $P\subset\mathbb{E}^{d}$ be a convex polytope with vertices
$p_{1},p_{2},\ldots,p_{n}$. Assume that for some hyperplane
$H^{\prime}\subset\mathbb{E}^{d}$ intersecting $[p_{1},p_{2}]$, we have
$p_{3},\ldots,p_{n}\in H^{\prime}$. Let $H$ be the hyperplane bisecting
$[p_{1},p_{2}]$. Let $\pi:\mathbb{E}^{d}\to H$ denote the orthogonal
projection onto $H$ and let $\sigma$ denote the Steiner symmetrization with
respect to $H$. Then

(1)
$\sigma(P)=\operatorname{conv}([p_{1},p_{2}]\cup\\{\pi(p_{3}),\ldots,\pi(p_{n})\\}).$

Furthermore,

(2) $\operatorname{ir}(\sigma(P))\geq\operatorname{ir}(P)$

with equality if and only if $P$ is symmetric to $H$. Finally, if for some
$1\leq k\leq d-1$, $P$ satisfies the property that for any $3\leq i_{1}\leq
i_{2}\leq\ldots\leq i_{m}$,
$\operatorname{conv}\\{p_{1},p_{i_{1}},\ldots,p_{i_{m}}\\}$ is a $k$-face of
$P$ if and only if $\operatorname{conv}\\{p_{2},p_{i_{1}},\ldots,p_{i_{m}}\\}$
is a $k$-face of $P$, then

(3)
$\operatorname{vol}_{k}(\operatorname{skel}_{k}(\sigma(P))\leq\operatorname{vol}_{k}(\operatorname{skel}_{k}(P)).$

For the proof of Lemma 1 we need Lemma 2. Lemma 2 explores the properties of
the so-called _linear parameter systems_ or _shadow systems_ , introduced by
Rogers and Shephard in [20]. Lemma 2 can be found in [13], and hence, we omit
its proof.

######  Lemma 2.

Let $p_{1},\ldots,p_{k},v\in\mathbb{E}^{d}$, 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 of Lemma 1.

Clearly, to prove (1), it is sufficient to prove the first equality, which is
the immediate consequence of Remark 4. The inequality in (2) follows from the
fact that the Steiner symmetral of a ball of radius $r$ is a ball of radius
$r$. The equality case in (2) is easy to see if $P$ is a triangle in
$\mathbb{E}^{d}$, and applying this observation for the sections of $P$ with
planes through $[p_{1},p_{2}]$ yields the statement in the general case.

Now we prove (3). Assume that $P$ is not symmetric to $H$. Let $v$ be a unit
normal vector of $H$, and for any $p_{i}$, where $i\geq 3$, let $d_{i}$ denote
the signed distance of $p_{i}$ from $H$ in such a way that $v$ points towards
the positive side of $H$. We define the linear parameter system
$L(t)=\\{p_{i}(t):i=1,2,\ldots,n\\}$, where $p_{1}(t)=p_{1}$, $p_{2}(t)=p_{2}$
and for $i=3,4,\ldots,n$, $p_{i}(t)=p_{i}-(1-t)d_{i}v$, and set
$P(t)=\operatorname{conv}(L(t))$. Observe that $P(1)=P$, $P(-1)$ is the
reflected copy of $P$ about $H$, and
$P(0)=\operatorname{conv}([p_{1},p_{2}]\cup\\{\pi(p_{i}),i=3,4,\ldots n\\}$ is
the Steiner symmetral of $P$ with respect to $H$. Note that for any value of
$t$, the points $p_{i}(t)$, where $i=3,4,\ldots,n$, lie in a rotated copy of
$H^{\prime}$ around $H\cap H^{\prime}$. Thus, by Lemma 2, the function
$g:[-1,1]\to\mathbb{R}$,
$g(t)=\operatorname{vol}_{k}(\operatorname{skel}_{k}(P(t)))$ is strictly
convex. Since $g(-t)=g(t)$ for any $t\in[-1,1]$, it implies that the unique
minimum of $g$ is attained at $t=0$. This readily yields (3). ∎

###  2.2. Hyperbolic Steiner symmetrization

In the following definition, for a hyperbolic line $L\subset\mathbb{H}^{d}$,
the line $L$ and the hypercycles with axis $L$ that are contained in a plane
through $L$ are called the _$g$ -lines of $\mathbb{H}^{d}$ with axis $L$_ (see
[18]).

######  Definition 2.

Let $H$ be a hyperplane in $\mathbb{H}^{d}$, and $L$ be a line orthogonal to
$H$. The _Steiner symmetrization $\sigma$_ with respect to $H$ and with axis
$L$ is the geometric transformation which assigns to any convex body
$K\subset\mathbb{H}^{d}$ the unique compact set $K^{\prime}$ symmetric to $H$
with the property that for any $g$-line $L^{\prime}$ with axis $L$, $K\cap
L^{\prime}$ and $K^{\prime}\cap L^{\prime}$ are both nondegenerate segments of
equal length, or singletons, or the empty set (see Figure 2). In this case we
call the set $\sigma(K)$ the _Steiner symmetral_ of $K$.

Figure 2. Hyperbolic Steiner symmetrization of a triangle $K$ with respect to
a horizontal line $H$ and axis $L$ in the projective disk model. Dotted lines
denote $g$-lines with axis $L$.

We note that hyperbolic Steiner symmetrization preserves volume [18,
Proposition 8].

In the next lemmas, if $H$ is a hyperplane in $\mathbb{H}^{d}$ orthogonal to a
line $L$, the _$g$ -orthogonal projection $\pi:\mathbb{H}^{d}\to H$ onto $H$
with axis $L$_ is defined by $\pi(p)=q$ for any $p\in\mathbb{H}^{d}$, where
$q$ is the intersection point of $H$ and the unique $g$-line through $p$ with
axis $L$. Our next lemma is proved by Peyerimhoff [18, Proposition 9], and
hence, we omit its proof.

######  Lemma 3.

Let $T=\operatorname{conv}\\{p_{1},p_{2},p_{3}\\}$ be a triangle in
$\mathbb{H}^{2}$. Let $L$ be the line through $[p_{1},p_{2}]$, and let $H$ be
the bisector of $[p_{1},p_{2}]$. Let $\pi:\mathbb{H}^{2}\to H$ denote the
$g$-orthogonal projection onto $H$ with axis $L$, and let $\sigma$ denote the
Steiner symmetrization with respect to $H$ and with axis $L$. Then

(4) $\sigma(T)\subseteq\operatorname{conv}\\{p_{1},p_{2},\pi(p_{3})\\},\hbox{
and}$ (5) $d(p_{1},p_{3})+d(p_{2},p_{3})\geq
d(p_{1},\pi(p_{3}))+d(p_{2},\pi(p_{3})).$

Furthermore, in any of (4) or (5), we have equality if and only if $p_{3}\in
H$.

######  Lemma 4.

Let $P\subset\mathbb{H}^{d}$ be a convex polytope with vertices
$p_{1},p_{2},\ldots,p_{n}$. Assume that for some hyperplane
$H^{\prime}\subset\mathbb{H}^{d}$ intersecting $[p_{1},p_{2}]$, we have
$p_{3},\ldots,p_{n}\in H^{\prime}$. Let $L$ be the straight line through
$[p_{1},p_{2}]$ and let $H$ be the bisector of $[p_{1},p_{2}]$. Let
$\pi:\mathbb{H}^{d}\to H$ denote the $g$-orthogonal projection onto $H$ with
axis $L$, and let $\sigma$ be the Steiner symmetrization with respect to $H$
with axis $L$. Then

(6)
$\sigma(P)\subseteq\operatorname{conv}([p_{1},p_{2}]\cup\\{\pi(p_{3}),\ldots,\pi(p_{n})\\}),$
(7)
$\operatorname{vol}_{d}(P)=\operatorname{vol}_{d}(\sigma(P))\leq\operatorname{vol}_{d}(\operatorname{conv}([p_{1},p_{2}]\cup\\{\pi(p_{3}),\ldots,\pi(p_{n})\\}),\hbox{
and}$ (8)
$\operatorname{ir}(P)\leq\operatorname{ir}(\operatorname{conv}([p_{1},p_{2}]\cup\\{\pi(p_{3}),\ldots,\pi(p_{n})\\})),$

with equality in any of (6), (7), or (8) if and only if $P$ is symmetric to
$H$.

###### Proof.

Note that $P=\bigcup_{F}(F\cap P)$, where $F$ runs over the planes containing
$L$, and these intersections are mutually disjoint apart from $[p_{1},p_{2}]$.
Let $F$ be a plane with $L\subset P$. By Remark 4, $F\cap
P=\operatorname{conv}\\{p_{1},p_{2},q\\}$ for some relative boundary point $q$
of $\operatorname{conv}\\{p_{3},\ldots,p_{n}\\}$. By Lemma 3, $\sigma(F\cap
P)\subseteq\operatorname{conv}(\sigma([p_{1},p_{2}])\cup\pi(q))$. Thus, we
have
$\sigma(P)\subseteq\operatorname{conv}(\sigma([p_{1},p_{2}])\cup\pi(\operatorname{conv}\\{p_{3},\ldots,p_{n}\\})$.
On the other hand, the relation
$\pi(\operatorname{conv}(p_{3},\ldots,p_{n}))\subseteq\operatorname{conv}\\{\pi(p_{3}),\ldots,\pi(p_{n})$
follows from [18, Lemma 10], which states that if $X$ is a convex set in $H$
containing the intersection point of $H$ and $L$, then $\pi^{-1}(X)$ is
convex. Thus, we have

$\sigma(P)\subseteq\operatorname{conv}(\sigma([p_{1},p_{2}])\cup\\{\pi(p_{3}),\ldots,\pi(p_{n})\\}).$

Here, by Lemma 3, there is equality if and only if all $p_{i}$ with $i\geq 3$
are contained in $H$. The second inequality in Lemma 4 readily follows from
the first one and the fact that hyperbolic volume is invariant under Steiner
symmetrization [18]. The inequality in (8), together with the equality case,
follows from the fact that the Steiner symmetral of a ball of radius $r$ is a
ball of radius $r$ [18, Proposition 8]. ∎

###  2.3. Spherical Steiner symmetrization

######  Definition 3.

Let $H$ be a hyperplane in $\mathbb{S}^{d}$. The _Steiner symmetrization
$\sigma$_ with respect to $H$ is the geometric transformation which assigns to
any convex body $K\subset\mathbb{S}^{d}$, disjoint from the two poles of $H$,
the unique compact set $K^{\prime}$ symmetric to $H$ with the property that
for any open half circle $L$ orthogonal to $H$ and ending at the two poles of
$H$, $K\cap L$ and $K^{\prime}\cap L$ are both nondegenerate segments of equal
length, or singletons, or the empty set (see Figure 3). In this case we call
the set $\sigma(K)$ the _Steiner symmetral_ of $K$.

Figure 3. Spherical Steiner symmetrization of a triangle $K$ with respect to
the spherical line $H$. Dotted lines denote $H$ and half circles perpendicular
to it.

We observe that a convex body $K$ disjoint from the poles of $H$ is contained
in an open hemisphere whose center lies in $H$; by our definition $\sigma(K)$
is contained in the same open hemisphere. Our next remark is the spherical
counterpart of [18, Lemma 10].

######  Remark 5.

Let $H$ be a hyperplane in $\mathbb{S}^{d}$ with poles $\pm p$, and let
$\pi:\mathbb{S}^{d}\setminus\\{\pm p\\}\to H$ denote the orthogonal projection
onto $H$. Then for any convex set $K\subset H$, the set $\pi^{-1}(K)$ is
convex.

The proof of our next lemma can be found in [4].

######  Lemma 5.

Let $H$ be the bisector of the segment $[p_{1},p_{2}]\in\mathbb{S}^{2}$, and
let $p_{3}$ be different from the poles of $H$. Let $\pi$ and $\sigma$ denote
the orthogonal projection onto $H$ and the Steiner symmetrization with respect
to $H$, respectively. Then

$\sigma(\operatorname{conv}\\{p_{1},p_{2},p_{3}\\})\subseteq\operatorname{conv}\\{p_{1},p_{2},\pi(p_{3})\\},$

with equality if and only if $p_{3}\in H$.

######  Lemma 6.

Let $P\subset\mathbb{S}^{d}$ be a convex polytope with vertices
$p_{1},p_{2},\ldots,p_{n}$. Assume that for some hyperplane
$H^{\prime}\subset\mathbb{S}^{d}$ intersecting $[p_{1},p_{2}]$, we have
$p_{3},\ldots,p_{n}\in H^{\prime}$. Let $H$ be the bisector of
$[p_{1},p_{2}]$, and assume that $P$ is disjoint from the poles of $H$. Let
$\pi:\mathbb{S}^{d}\to H$ denote the orthogonal projection onto $H$, and let
$\sigma$ be the Steiner symmetrization with respect to $H$. Then

(9)
$\sigma(P)\subseteq\operatorname{conv}([p_{1},p_{2}]\cup\\{\pi(p_{3}),\ldots,\pi(p_{n})\\}),\hbox{
and}$ (10)
$\operatorname{vol}_{d}(P)\leq\operatorname{vol}_{d}(\sigma(P))\leq\operatorname{vol}_{d}(\operatorname{conv}([p_{1},p_{2}]\cup\\{\pi(p_{3}),\ldots,\pi(p_{n})\\}))$

with equality in any of (9) and (10) if and only if $P$ is symmetric to $H$.

###### Proof.

By Remark 4 and Lemma 5,
$\sigma(P)\subseteq\operatorname{conv}([p_{1},p_{2}]\cup\pi(P))$. On the other
hand, by Remark 5,
$\pi(P)\subseteq\operatorname{conv}\\{q,\pi_{H}(p_{3}),\ldots,\pi_{H}(p_{n})\\}$,
where $q$ is the midpoint of $[p_{1},p_{2}]$. Thus, we have
$\sigma(P)\subseteq\operatorname{conv}([p_{1},p_{2}]\cup\\{\pi(p_{3}),\ldots,\pi(p_{n})\\})$,
where, by Lemma 5, we have equality if and only if $p_{3},\ldots,p_{n}\subset
H$, that is, if $P$ is symmetric to $H$. This, and the fact that spherical
volume does not decrease under Steiner symmetrization implies (10). ∎

##  3\. Main results

We start with Theorem 1, which was proved for $\mathbb{M}=\mathbb{E}^{d}$ in
[10].

######  Theorem 1.

Let $\mathbb{M}\in\\{\mathbb{E}^{d},\mathbb{H}^{d},\mathbb{S}^{d}\\}$. Let
$B\subset\mathbb{M}$ be a ball, and let $P\subset B$ be a convex polytope with
$d+2$ vertices. Then

$\operatorname{vol}_{d}(P)\leq\operatorname{vol}_{d}(Q),$

where $Q=\operatorname{conv}(S_{1}\cup S_{2})$ for some regular simplices
$S_{1},S_{2}$ inscribed in $B$, with $\dim S_{1}=k$ and $\dim S_{2}=d-k$ for
some $1\leq k\leq\left\lfloor\frac{d}{2}\right\rfloor$, and contained in
mutually orthogonal subspaces of $\mathbb{M}$ intersecting at the center of
$B$. Furthermore, for any $d>0$ there is a value $\varepsilon>0$ such that if
the radius of $B$ is at most $\varepsilon$, or $\mathbb{M}=\mathbb{E}^{d}$,
then $k=\left\lfloor\frac{d}{2}\right\rfloor$.

###### Proof.

By compactness argument, in the family of convex polytopes $P\subset B$ with
at most $d+2$ vertices, $\operatorname{vol}_{d}(\cdot)$ attains its maximum,
and any polytope with maximal volume has exactly $d+2$ vertices. Let $P$ be
such a polytope. Then, by Remark 1, $P=\operatorname{conv}(S_{1}\cup
S_{2})\subset B$ for some simplices $S_{1},S_{2}$ of dimensions $k$ and $d-k$,
respectively, where $1\leq k\leq\left\lfloor\frac{d}{2}\right\rfloor$, and the
subspaces spanned by them intersect in a single common point $p$ of $S_{1}$
and $S_{2}$. Let $L_{1}$ (resp. $L_{2}$) denote the $k$-dimensional (resp.
$(d-k)$-dimensional) subspace spanned by $S_{1}$ (resp. $S_{2}$). Note that
without loss of generality, we may assume that every vertex of $P$ lies on the
boundary of $B$.

Consider an edge $[q_{1},q_{2}]$ of $S_{1}$. We show that $P$ is symmetric to
the bisector $H$ of $[q_{1},q_{2}]$.

Suppose for contradiction that $P$ is not symmetric to $H$. Let us choose a
hyperplane $H^{\prime}$ containing $V(P)\setminus\\{q_{1},q_{2}\\}$ such that
$H^{\prime}$ intersects $[q_{1},q_{2}]$, and recall that by Remark 3, such a
hyperplane exists. Let $L$ denote the line through $[q_{1},q_{2}]$. Note that
since $P$ is inscribed in $B$, the bisector $H$ of $[q_{1},q_{2}]$ passes
through the center of $B$. Let $\sigma$ denote the Steiner symmetrization with
respect to $H$ (or if $\mathbb{M}=\mathbb{H}^{d}$, with respect to $H$ and
with axis $L$), and let $\pi$ denote the orthogonal projection onto $H$.

Let
$P^{\prime}=\operatorname{conv}([q_{1},q_{2}]\cup\pi(V(P)\setminus\\{q_{1},q_{2}\\}))$.
Then, for $\mathbb{M}=\mathbb{S}^{d}$ and $\mathbb{M}=\mathbb{H}^{d}$, by
Lemmas 6 and 4, respectively, we have that
$\operatorname{vol}_{d}(P)<\operatorname{vol}_{d}(P^{\prime})$, contradicting
our assumption that $P$ has maximal volume. In the remaining case, if
$\mathbb{M}=\mathbb{E}^{d}$, then $P^{\prime}$ is a convex polytope in $B$
with $(d+2)$ vertices satisfying
$\operatorname{vol}_{d}(P^{\prime})=\operatorname{vol}_{d}(P)$ such that at
least one vertex of $P^{\prime}$ lies in the interior of $B$. Thus, there is a
convex polytope $P^{\prime\prime}\subset B$ with $(d+2)$ vertices and
satisfying
$\operatorname{vol}_{d}(P^{\prime\prime})>\operatorname{vol}_{d}(P)$; a
contradiction.

We have shown that $P$ is symmetric to the bisector of the edge
$[q_{1},q_{2}]$ of $S_{1}$. We obtain similarly that $P$ is symmetric to any
edge of $S_{1}$ and $S_{2}$, implying that $S_{1}$ and $S_{2}$ are regular
simplices inscribed in $B$ contained in orthogonal subspaces such that the
centers of $S_{1}$, $S_{2}$ and $B$ coincide.

The fact that if $\mathbb{M}=\mathbb{E}^{d}$, the volume of
$\operatorname{conv}(S_{1}\cup S_{2})$ is maximal if
$k=\left\lfloor\frac{d}{2}\right\rfloor$ follows by a simple computation (see,
e.g. [10]). The existence of some $\varepsilon>0$ depending on $d$ only, such
that for any ball $B$ of radius at most $\varepsilon$, the volume of
$\operatorname{conv}(S_{1}\cup S_{2})$ is maximal if
$k=\left\lfloor\frac{d}{2}\right\rfloor$ follows from the well known
observation that in ‘small scale’, hyperbolic and spherical spaces are
‘almost’ Euclidean, and can be proved by a standard limit argument. ∎

Our next two theorems deal with the volumes of polytopes in Euclidean and
hyperbolic spaces, containing a given ball.

######  Theorem 2.

Let $\mathbb{M}\in\\{\mathbb{E}^{d},\mathbb{H}^{d}\\}$ and let
$B\subset\mathbb{M}$ be a ball. Then, among simplices containing $B$, the ones
with maximal volume are the regular simplices circumscribed about $B$.

######  Theorem 3.

Let $\mathbb{M}\in\\{\mathbb{E}^{d},\mathbb{H}^{d}\\}$, let
$B\subset\mathbb{M}$ be a ball, and let $P$ be a convex polytope containing
$B$ with at most $d+2$ vertices. Then

$\operatorname{vol}_{d}(P)\geq\operatorname{vol}_{d}(Q),$

for a convex polytope $Q=\operatorname{conv}(S_{1}\cup S_{2})$ containing $B$,
where $S_{1},S_{2}$ are regular simplices with $\dim S_{1}=k$ and $\dim
S_{2}=d-k$ for some $1\leq k\leq\frac{d}{2}$, and contained in mutually
orthogonal subspaces of $\mathbb{M}$ intersecting at the center of $B$.
Furthermore, for any $d>0$ there is a value $\varepsilon>0$ such that if the
radius of $B$ is at most $\varepsilon$, or $\mathbb{M}=\mathbb{E}^{d}$, then
$k=\left\lfloor\frac{d}{2}\right\rfloor$, or $d=4$ and $k=1$.

We note that while the natural choice to solve isoperimetric problems in
$\mathbb{H}^{d}$ or $\mathbb{S}^{d}$ is to use hyperbolic or spherical Steiner
symmetrization, sometimes Euclidean Steiner symmetrization can also be used in
these spaces. We illustrate it in the proof of Theorems 2 and 3 in hyperbolic
space. The proof presented in this paper is a variant of the proof of Lemma 9
in [2].

###### Proof of Theorems 2 and 3.

We start with the proof of Theorem 2.

First, assume that $\mathbb{M}=\mathbb{H}^{d}$. We represent $\mathbb{H}^{d}$
in the projective ball model. As we already remarked in Section 2, in this
model the points of $\mathbb{H}^{d}$ correspond to the points of the open unit
ball $\operatorname{int}(\mathbf{B}^{d})$ of $\mathbb{E}^{d}$ centered at the
origin, and a set is convex in $\mathbb{H}^{d}$ if and only if it is
represented by a convex set in $\operatorname{int}(\mathbf{B}^{d})$. The
volume element at point $x$ in this model is
$\frac{1}{\left(1+||x||^{2}\right)^{\frac{d+1}{2}}}dx$. Hence, if $A$ is a
Borel set in $\operatorname{int}(\mathbf{B}^{d})$, then

$\operatorname{vol}(A)=\int_{A}\frac{1}{\left(1+||x||^{2}\right)^{\frac{d+1}{2}}}dx,$

where $dx$ denotes integration with respect to $d$-dimensional Lebesgue
measure [19].

Let $S$ be a simplex in $\mathbb{H}^{d}$ containing $B$. Without loss of
generality, we may assume that the center of $B$ is $o$. Observe that by
compactness, $\operatorname{vol}_{d}(\cdot)$ attains its minimum on the family
of hyperbolic simplices in $\mathbb{H}^{d}$. Let $S$ be a simplex in $B$
minimizing this volume, and let $[p_{1},p_{2}]$ be an edge of $S$. Let $H$
denote the hyperplane perpendicular to the line through $[p_{1},p_{2}]$ that
passes through $o$. Let $\sigma$ denote the (Euclidean) Steiner symmetrization
with respect to $H$. Then, clearly, $\sigma(S)$ is a hyperbolic simplex in
$\mathbb{H}^{d}$ containing $B$. Furthermore, since the density function
$\frac{1}{\left(1+||x||^{2}\right)^{\frac{d+1}{2}}}$ is a strictly increasing
function of $||x||$, by Fubini’s theorem we have that

$\operatorname{vol}_{d}(\sigma(S))=\int_{\sigma(S)}\frac{1}{\left(1+||x||^{2}\right)^{\frac{d+1}{2}}}dx\leq\int_{S}\frac{1}{\left(1+||x||^{2}\right)^{\frac{d+1}{2}}}dx=\operatorname{vol}_{d}(S),$

with equality if and only if $\sigma(S)=S$, that is, if $S$ is symmetric to
$H$. Since $S$ has a minimal hyperbolic volume on the family of simplices
containing $B$, it follows that the bisector of any edge of $S$ contains $o$,
and thus, $S$ is a regular simplex with $o$ as its center. We obtain also that
$S$ is circumscribed about $B$, as otherwise $\operatorname{vol}_{d}(S)$ is
clearly not minimal. This proves Theorem 2 for $\mathbb{M}=\mathbb{H}^{d}$.
For $\mathbb{M}=\mathbb{E}^{d}$, and to obtain the inequality in Theorem 3, we
can apply an analogous argument.

Now we investigate the equality part of Theorem 3. By a standard limit
argument, it is sufficient to prove the statement for
$\mathbb{M}=\mathbb{E}^{d}$, where we may assume that $B$ is the unit ball
centered at the origin.

The statement clearly holds if $d=2$, and hence, we assume that $d\geq 3$. Let
$P$ be a convex polytope in $\mathbb{E}^{d}$ with $B\subset P$ and minimal
volume. We show that then

(11)
$\operatorname{vol}_{d}(P)\leq\frac{d^{d/2}}{d!}\left(\left\lfloor\frac{d}{2}\right\rfloor\right)^{(\lfloor
d/2\rfloor+1)/2}\left(\left\lceil\frac{d}{2}\right\rceil\right)^{(\lceil
d/2\rceil+1)/2}\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right).$

First, an elementary computation shows that the volume of a regular simplex in
$\mathbb{E}^{d}$ with unit inradius is $\frac{d^{d/2}(d+1)^{(d+1)/2}}{d!}$,
which is strictly smaller than the quantity in (11). Thus, in the following we
consider the case that $P$ has exactly $d+2$ vertices.

Similarly like in the proof of Theorem 2, we have that if $P$ has minimal
volume, then $P=\operatorname{conv}(S_{1}\cup S_{2})$ for two regular
simplices $S_{1}$ and $S_{2}$, where the affine hulls of $S_{1}$ and $S_{2}$
are perpendicular and meet at the common centerpoint of $S_{1}$ and $S_{2}$.
Let $1\leq\dim S_{1}=k\leq\dim S_{2}=d-k\leq d-1$. Without loss of generality,
we may assume that $o$ is the center of $S_{1}$ and $S_{2}$.

We determine the edge lengths and the dimensions of $S_{1}$ and $S_{2}$. For
$i=1,2$, let the inradius of $S_{i}$ be denoted by $r_{i}$. Let the vertices
of $S_{1}$ and $p_{1},\ldots,p_{k+1}$, and the vertices of $S_{2}$ be
$q_{1},\ldots,q_{d-k+1}$. Note that for any value of $i$, $||p_{i}||=kr_{1}$,
and $||q_{i}||=(d-k)r_{2}$. Furthermore, since a facet of $P$ is the convex
hull of a facet of $S_{1}$ and a facet of $S_{2}$, the volume of $P$ can be
obtained as $(k+1)(d-k+1)$ times the volume of
$\operatorname{conv}\\{o,p_{1},\ldots,p_{k},q_{1},\ldots,q_{d-k}\\}$. In
addition, the latter volume is $d!$ times the volume of the parallelotope
$P^{\prime}$ spanned by the vectors $p_{1},\ldots,p_{k},q_{1},\ldots,q_{d-k}$.
Let $G$ denote the Gram matrix of these vectors, and note that for any values
of $i,j$, $\langle p_{i},p_{i}\rangle=k^{2}r_{1}^{2}$, $\langle
q_{i},q_{i}\rangle=(d-k)^{2}r_{2}^{2}$ and $\langle p_{i},q_{j}\rangle=0$, and
for any $i\neq j$, $\langle p_{i},p_{j}\rangle=-kr_{1}^{2}$ and $\langle
q_{i},q_{j}\rangle=-(d-k)r_{2}^{2}$. It is well known that
$\operatorname{vol}_{d}(P^{\prime})=\sqrt{\det(G)}$. Thus, an elementary
computation shows that
$\operatorname{vol}_{d}(P^{\prime})=r_{1}^{k}r_{2}^{d-k}k^{k+1/2}(d-k)^{d-k+1/2}$,
which yields that

(12)
$\operatorname{vol}_{d}(P)=\frac{1}{d!}r_{1}^{k}r_{2}^{d-k}k^{k+1/2}(d-k)^{d-k+1/2}(k+1)(d-k+1).$

Note that since the unit ball centered at $o$ touches at least one facet of
$P$, it touches every facet of $P$. Let $c_{1}$ and $c_{2}$ be the centers of
a facet of $S_{1}$ and a facet of $S_{2}$, respectively. Then the unit ball
centered at $o$ touches the segment $[c_{1},c_{2}]$. Since the triangle
$\operatorname{conv}\\{o,c_{1},c_{2}\\}$ is a right triangle with legs of
lengths $r_{1}$ and $r_{2}$, respectively, we obtain that
$(r_{1}^{2}-1)(r_{2}^{2}-1)=1$. Thus, there is a real value $t\in\mathbb{R}$
such that $r_{1}=\sqrt{1+e^{t}}$ and $r_{2}=\sqrt{1+e^{-t}}$. Substituting
these expressions into (12), we obtain that
$\operatorname{vol}_{d}(P)=f_{k}(t)$, where

$f_{k}(t)=\frac{1}{d!}(1+e^{t})^{k/2}(1+e^{-t})^{(d-k)/2}k^{k+1/2}(d-k)^{d-k+1/2}(k+1)(d-k+1).$

We find the minimum of $f_{k}(t)$ for all real $t\in\mathbb{R}$ and integer
$1\leq k\leq d-1$. Observe that for any value of $k$, $f_{k}(t)$ attains its
minimum for some value of $t$. On the other hand, an elementary computation
shows that the only real solution of the equation $f^{\prime}_{k}(t)=0$ is
$t=\frac{d-k}{k}$, which shows that for any fixed value of $k$, $f_{k}(t)$ is
minimal if $t=\frac{d-k}{k}$. Let $g_{d}(k)=f_{k}\left(\frac{d-k}{k}\right)$.
Then

$g_{d}(k)=\frac{d^{d/2}}{d!}k^{(k+1)/2}(d-k)^{(d-k+1)/2}(k+1)(d-k+1).$

We remark that for $d=3$ we have proved the statement, and hence, we assume
that $d\geq 4$. Let us extend the domain of $g_{d}$ to the interval
$k\in[1,d-1]$ using the same formula. Then by differentiating twice and
applying algebraic transformations, we obtain that the function $\ln g_{d}$ is
strictly convex on $[2,d-2]$, implying that for any integer $2\leq
k<\lfloor\frac{d}{2}\rfloor$, we have
$g_{d}(k)>g_{d}\left(\lfloor\frac{d}{2}\rfloor\right)$. On the other hand, an
elementary computation shows that $g_{d}(1)\geq g_{d}(2)$, with equality if
and only if $d=4$. This yields the assertion. ∎

######  Corollary 1.

Let $P$ be a convex polytope in $\mathbb{E}^{d}$ with at most $d+2$ vertices
and unit inradius. Then

$\operatorname{vol}_{d}(P)\leq\frac{d^{d/2}}{d!}\left(\left\lfloor\frac{d}{2}\right\rfloor\right)^{(\lfloor
d/2\rfloor+1)/2}\left(\left\lceil\frac{d}{2}\right\rceil\right)^{(\lceil
d/2\rceil+1)/2}\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)\left(\left\lceil\frac{d}{2}\right\rceil+1\right),$

with equality if and only if there are regular simplices $S_{1},S_{2}$ lying
in orthogonal affine subspaces that meet at the common center of both $S_{1}$
and $S_{2}$, such that one of the following holds:

  * •

$\dim S_{1}=\left\lfloor\frac{d}{2}\right\rfloor$ and $\dim
S_{2}=\left\lceil\frac{d}{2}\right\rceil$;

  * •

$d=4$, $\dim S_{1}=1$ and $\dim S_{2}=3$.

We are unable to extend Theorem 2 for $\mathbb{M}=\mathbb{S}^{d}$.
Nevertheless, we note that the spherical version of Dowker’s theorem implies
that if $\mathbb{M}=\mathbb{S}^{2}$, then any spherical $m$-gon of minimal
area containing the spherical cap $B$ is a regular $m$-gon circumscribed about
$P$. We present a proof for $\mathbb{M}=\mathbb{S}^{3}$ using László Fejes
Tóth’s famous ‘moment lemma’, which we quote as follows (see [8]).

######  Lemma 7 (Moment lemma).

Let $\rho(\tau)$ be a strictly increasing function for $0<\tau<\frac{\pi}{2}$.
Furthermore, let $p_{1},p_{2},\ldots,p_{k}$ be $k$ points of $\mathbb{S}^{2}$
not all of them lying on a hemisphere, and let $\tau(p)$ denote the minimal
spherical distance of the point $p\in\mathbb{S}^{2}$ from a point of the set
$\\{p_{1},p_{2},\ldots,p_{k}\\}$. Let $\bar{\Delta}$ be an equilateral
spherical triangle with vertices $\bar{p}_{1},\bar{p}_{2},\bar{p}_{3}$ and
having area $\frac{2\pi}{k-2}$, and let $\bar{\tau}(p)$ denote the minimum
spherical distance of $p$ from one of the vertices of $\bar{\Delta}$. Then we
have

(13)
$\int_{\mathbb{S}^{2}}\rho(\tau(p))\,d\omega_{s}\leq(2k-4)\int_{\bar{\Delta}}\rho(\bar{\tau}(p))\,d\omega_{s},$

where $d\omega_{s}$ denotes integration with respect to spherical volume.
Equality in 13 holds if and only if $\\{p_{i}:i=1,2,\ldots,k\\}$ is the vertex
set of a regular tetrahedron, octahedron or icosahedron.

######  Theorem 4.

Let $B\subset\mathbb{S}^{3}$ be a spherical ball. Let $T\subset\mathbb{S}^{3}$
be a tetrahedron containing $B$, and let $T_{\mathrm{reg}}$ be a regular
tetrahedron circumscribed about $B$. Then

$\operatorname{vol}_{3}(T)\geq\operatorname{vol}_{3}(T_{\mathrm{reg}}),$

with equality if and only if $T$ is congruent to $T_{\mathrm{reg}}$.

###### Proof.

Clearly, we may assume that $T$ is circumscribed about $B$. Let $p_{i}$, where
$i=1,2,3,4$, denote the points where a face of $T$ touches $B$. Observe that
$\operatorname{bd}(B)$ is a $2$-dimensional spherical space, and that the
central projections of the faces of $T$ onto $\operatorname{bd}(B)$ coincide
with the Voronoi cells of the point set $\\{p_{1},p_{2},p_{3},p_{4}\\}$; i.e.
any point of the cell containing $p_{i}$ is not farther from $p_{i}$ than from
any other point of the set. Furthermore, if for any
$p\in\operatorname{bd}(B)$, $\tau(p)$ denotes the minimum spherical distance
from the points $p_{i}$, then there is a strictly increasing function
$\rho(\tau)$ such that the volume of $T$ coincides with
$\int_{\operatorname{bd}(B)}\rho(\tau(p))\,d\omega_{s}$. Thus, the assertion
follows from Lemma 7. ∎

Is is well known (see e.g. [14]) that among simplices $S$ in $\mathbb{E}^{d}$,
the ratio of the circumradius of $S$ to its inradius is minimal if and only if
$S$ is regular. Our next result is a partial generalization of this for
hyperbolic and spherical spaces.

######  Theorem 5.

Let $d\geq 2$. For any simplex $S$ in the hyperbolic space $\mathbb{H}^{d}$,

$\tanh\operatorname{cr}(S)\geq d\tanh\operatorname{ir}(S),$

with equality if, and only if $S$ is regular. Furthermore, for any simplex $S$
in the spherical space $\mathbb{S}^{d}$ with $d\leq 3$,

$\tan\operatorname{cr}(S)\geq d\tan\operatorname{ir}(S),$

with equality if, and only if $S$ is regular.

###### Proof.

Among simplices $S$ of a given volume in $\mathbb{H}^{d}$ or $\mathbb{S}^{d}$,
the regular ones have minimal circumradius (see [18] and [4], respectively).
Furthermore, among simplices of a given volume in $\mathbb{H}^{d}$, the
regular ones have maximal inradius (see Theorem 2), and the same holds for
$d\leq 3$ for simplices in $\mathbb{S}^{d}$ (see Theorem 4, and the remark
preceding it). Theorem 5 readily follows from these observations. ∎

We remark that the $d=2$ case of Theorem 5 can be found e.g. in [22]. Before
stating our next result, we recall that by [18], among simplices in
$\mathbb{H}^{d}$ containing a given ball $B$ the regular simplices
circumscribed about $B$ have minimal total edge length. In our next statement
we show that there is no similar statement for spherical simplices, and prove
a variant of this statement for hyperbolic polytopes with $d+2$ vertices.

######  Proposition 1.

Let $d\geq 2$, let $B$ be a ball of radius $r$ in $\mathbb{S}^{d}$, and let
$S_{\mathrm{reg}}$ be a regular simplex circumscribed about $B$. We have the
following.

  * (i)

For any $d\geq 3$, if $r$ is sufficiently close to $\frac{\pi}{2}$, then there
is a simplex $S$ containing $B$ and satisfying
$\operatorname{vol}_{1}(\operatorname{skel}_{1}(S))<\operatorname{vol}_{1}(\operatorname{skel}_{1}(S_{\mathrm{reg}}))$.

  * (ii)

For any $0<r<\frac{\pi}{2}$, if $d$ is sufficiently large, then there is a
simplex $S$ containing $B$ and satisfying
$\operatorname{vol}_{1}(\operatorname{skel}_{1}(S))<\operatorname{vol}_{1}(\operatorname{skel}_{1}(S_{\mathrm{reg}}))$.

###### Proof.

First, we compute
$\operatorname{vol}_{1}(\operatorname{skel}_{1}(S_{\mathrm{reg}}))$. Let $\pi$
denote the central projection onto the tangent hyperplane $H$ of
$\mathbb{S}^{d}$ in $\mathbb{E}^{d+1}$ at the center of $S_{\mathrm{reg}}$,
and note that $S_{\mathrm{reg}}$ is contained in the open hemisphere centered
at this point. Thus, $\pi(S_{\mathrm{reg}})$ is a regular Euclidean simplex
with inradius $\tan r$, implying that its circumradius is $d\tan r$. Regarding
the center $c$ of $S_{\mathrm{reg}}$ as the origin of the Euclidean $d$-space
$H$ and taking the inner product of two vertices of $\pi(S_{\mathrm{reg}})$,
we have that if $p_{i}$ and $p_{j}$ are two vertices of $S_{\mathrm{reg}}$,
then the angle of the Euclidean triangle with vertices $c$, $\pi(p_{i})$ and
$\pi(p_{j})$ at the vertex $c$ is $\arccos\left(-\frac{1}{d}\right)$. This
quantity is equal to the angle of the sperical triangle with vertices
$p_{i},p_{j}$ and $c$ at $c$. Since the length of the edges $[c,p_{i}]$ and
$[c,p_{j}]$ in this triangle is $\arctan(d\tan r)$, by the spherical Law of
Cosines and using trigonometric identities we obtain that the length of the
edges of $S_{\mathrm{reg}}$ is $\arccos\frac{1-d\tan^{2}r}{1+d^{2}\tan^{2}r}$.
Thus, $\operatorname{vol}_{1}(\operatorname{skel}_{1}(S_{\mathrm{reg}}))$ is
equal to $f(r,d)=\binom{d+1}{2}\arccos\frac{1-d\tan^{2}r}{1+d^{2}\tan^{2}r}$.

We show that for any $\varepsilon>0$, there is a simplex $S$ containing $B$
with $\operatorname{vol}_{1}(\operatorname{skel}_{1}(S))<d\pi+\varepsilon$.
Let $S_{0}$ be a regular $(d-1)$-dimensional simplex whose total edge length
is strictly less than $\varepsilon$. We assume that the spherical segment
connecting the center $c^{\prime}$ of $S_{0}$ to $c$ is perpendicular to
$S_{0}$, and denote the length of this segment by $x$. Assume that
$x<\frac{\pi}{2}$, and let $p$ be a point on the line through this segment
such that $c$ is the midpoint of $[c^{\prime},p]$. Observe that if $x$ is
sufficiently close to $\frac{\pi}{2}$, then
$S=\operatorname{conv}(S_{0}\cup\\{p\\})$ contains $B$. Furthermore,
$\operatorname{vol}_{1}(\operatorname{skel}_{1}(S))<d\pi+\varepsilon$.

Thus, to show the statement it is sufficient to show that if $d\geq 3$ is
fixed and $\tan r$ is sufficiently large, or if $0<r<\frac{\pi}{2}$ is fixed
and $d$ is sufficiently large, then $f(d,r)-d\pi$ is positive, which can be
checked by an elementary computation. ∎

Before Theorems 6-8, we remark that unlike volume and surface area, the
quantity $\operatorname{vol}_{k}(\operatorname{skel}_{k}(P))$ is not
continuous in the family of $d$-dimensional convex polytopes, if $1\leq k\leq
d-2$.

######  Theorem 6.

Let $B\subset\mathbb{H}^{d}$ be a ball with center $c$, and let $P$ be a
convex polytope containing $B$ and with $V(P)\leq d+2$ such that $P$ has
minimal total edge length among the polytopes in $\mathbb{H}^{d}$ containing
$B$ and having at most $d+2$ vertices. Then we have one of the following.

  * (i)

There are regular simplices $S_{1}$ and $S_{2}$ with $\dim S_{1}+\dim S_{2}=d$
such that the subspaces spanned by $S_{1}$ and $S_{2}$ are orthogonal
complements of each other, $S_{1}\cap S_{2}=\\{c\\}$, and
$P=\operatorname{conv}(S_{1}\cup S_{2})$.

  * (ii)

There are perpendicular segments $S_{1},S_{2}$ meeting at the common midpoint
$q_{1}$ of both segments, and a regular $(d-2)$-dimensional simplex $Q$ with
center $q_{2}$ such that $c$ lies on the segment $[q_{1},q_{2}]$, $Q$ is
perpendicular to $[q_{1},q_{2}]$, and $P=\operatorname{conv}(S_{1}\cup
S_{2}\cup Q)$.

  * (iii)

$P$ is a regular simplex circumscribed about $B$.

###### Proof.

We note that if $V(P)=d+1$, then the statement follows from [18, Theorem 3].
Thus, we assume that $V(P)=d+2$. By Remark 1, there are simplices $S_{1}$ and
$S_{2}$ with $\dim S_{i}=k_{i}$ for $i=1,2$ and intersecting at a singleton
which is a relative interior point of both, and there are points
$p_{1},p_{2},\ldots,p_{r}$ such that $P=\operatorname{conv}(S_{1}\cup
S_{2}\cup\\{p_{1},\ldots,p_{r}\\})$, and $k_{1}+k_{2}+r=d$. If $r>0$, let
$Q=\operatorname{conv}\\{p_{1},\ldots,p_{r}\\}$.

Let $[q_{1},q_{2}]$ be any edge of $S_{1}$. Then there is a unique hyperplane
$H^{\prime}$ containing all other vertices of $P$ such that $H^{\prime}$
intersects $[q_{1},q_{2}]$ at a point $q$. Let $L$ be the line containing
$[q_{1},q_{2}]$, and let $H$ denote the bisector of $[q_{1},q_{2}]$. Let
$\sigma$ denote Steiner symmetrization with respect to $H$ with axis $L$. We
show that if $P$ is not symmetric to $H$, then
$\operatorname{conv}(\sigma(P))$ is a convex polytope with
$\operatorname{ir}(P)\leq\operatorname{ir}(\operatorname{conv}(\sigma(P)))$
and strictly smaller total edge length. Here, the first property follows from
(8). Furthermore, if $k_{2}>1$, then any pair of vertices not in $S_{1}$ are
connected by an edge, which, by Lemma 3 and the fact the length of a segment
does not increase under $g$-orthogonal projection, implies the second
property. Finally, assume that $k_{2}=1$, i.e $S_{2}=[w_{1},w_{2}]$. By Lemma
3, it is sufficient to show that $[\pi(w_{1}),\pi(w_{2})]$ is not an edge of
$\sigma(P)$, where $\pi:\mathbb{H}^{d}\to H$ denotes $g$-ortogonal projection
onto $H$ with axis $L$.

Observe that $\operatorname{conv}(V(P)\setminus\\{w_{1},w_{2}\\})$ intersects
$S_{2}$ in a relative interior point of $S_{2}$. Let this point be $w$. The
line $L^{\prime}\subset H^{\prime}$ through $q$ and $w$ intersects
$\operatorname{bd}(P)$ in $q$ and another point, which we denote by
$w^{\prime}$. Then, by [18, Lemma 10] (see also the proof of Lemma 4),
$\pi(\operatorname{conv}\\{w_{1},w_{2},q,w^{\prime}\\})$ is a nondegenerate
quadrangle in $H$ with $\operatorname{conv}\\{\pi(w_{1}),\pi(w_{2})\\}$ as a
diagonal. This, as $\pi(w_{1})$, $\pi(w_{2})$ are vertices of $\sigma(P)$,
yields also that the segment connecting them is not an edge of $\sigma(P)$,
and thus, if $P$ is not symmetric to $H$,
$\operatorname{vol}_{1}(\operatorname{skel}_{1}(\sigma(P)))<\operatorname{vol}_{1}(\operatorname{skel}_{1}(P))$.

We have shown that $P$ is symmetric to the bisector of every edge of $S_{1}$.
Note that if $r>0$ and $k_{1}>1$, then
$S^{\prime}=\operatorname{conv}(S_{1}\cup Q^{\prime})$ is a simplex
intersecting $S_{2}$ at a relative interior point of $S_{2}$, and thus, by the
same argument, $P$ is symmetric to the bisector of every edge of $S^{\prime}$.
Thus, if $k_{1}>1$ or $k_{2}>1$, then $r=0$, and the properties in (i) are
satisfied for $P$. Furthermore, if $k_{1}=k_{2}=1$, then we obtain similarly
that $P$ is symmetric to the bisector of $S_{1}$, $S_{2}$ and every edge of
$Q$, implying the properties in (ii). ∎

We note that the statement of Theorem 7 for simplices can be found in [13].

######  Theorem 7.

For any $1\leq k\leq d-1$, among simplices in $\mathbb{E}^{d}$ with unit
volume, the regular ones have minimal total $k$-content. Furthermore, for any
$1\leq k\leq d-1$, among the convex polytopes in $\mathbb{E}^{d}$ with at most
$d+2$ vertices and unit volume, if $P$ has minimal total $k$-content, then it
satisfies one of the following.

  * (i)

There are regular simplices $S_{1}$ and $S_{2}$ with $\dim S_{1}+\dim S_{2}=d$
such that the subspaces spanned by $S_{1}$ and $S_{2}$ are orthogonal
complements of each other, $S_{1}\cap S_{2}=\\{c\\}$, and
$P=\operatorname{conv}(S_{1}\cup S_{2})$.

  * (ii)

There are regular simplices $S_{1},S_{2}$, of dimensions $\dim S_{1}=k_{1}\leq
k$ and $\dim S_{2}=k_{2}\leq k$ and lying in orthogonal subspaces that meet at
the common center $q_{1}$ of both simplices, and a regular
$(d-k_{1}-k_{2})$-dimensional simplex $Q$ with center $q_{2}$ such that $Q$ is
perpendicular to the segment $[q_{1},q_{2}]$, and
$P=\operatorname{conv}(S_{1}\cup S_{2}\cup Q)$.

  * (iii)

$P$ is a regular simplex of unit volume.

###### Proof.

Let $P$ be a simplex of unit volume in $\mathbb{E}^{d}$, and let
$[p_{1},p_{2}]$ be an edge of $S$. Let $H$ denote the bisector of
$[p_{1},p_{2}]$, and let $\sigma$ denote Steiner symmetrization with respect
to $H$. If $P$ is not symmetric to $H$, then $\sigma(P)$ is a unit volume
simplex with strictly smaller total $k$-content. Thus, a simplex with minimal
total $k$-content is symmetric to the bisector of any of its edges, implying
that it is regular.

Now, let $P$ be a $d$-dimensional convex polytope of unit volume and having
exactly $d+2$ vertices. Then there are simplices $S_{1}$, $S_{2}$ of
dimensions $k_{1}$ and $k_{2}$ respectively, and possibly a simplex
$Q=\operatorname{conv}\\{p_{1},p_{2},\ldots,p_{r}\\}$ such that the affine
hulls of $S_{1}$ and $S_{2}$ intersect in a singleton $q_{1}$ which lies in
the relative interior of both $S_{1}$ and $S_{2}$, and $k_{1}+k_{2}+r=d$.

Consider the case that one of $k_{1}$ and $k_{2}$, say $k_{1}$, is greater
than $k$. Let $[p_{1},p_{2}]$ be an edge of $\operatorname{conv}(S_{1}\cup
Q)$. It is easy to see that any face of $P$ not containing
$\operatorname{conv}(S_{1}\cup S_{2})$ is a simplex. Thus, any $k$-face of $P$
containing at most one of $p_{1}$ and $p_{2}$ is a simplex. Furthermore, by
the properties of the Gale diagram of $P$ described in Remark 2, for any
$k$-element subset $S^{\prime}$ of the vertices of $P$ with $p_{1},p_{2}\notin
S^{\prime}$, we have that $\operatorname{conv}(\\{p_{1}\\}\cup S^{\prime})$ is
a $k$-face of $P$ if and only if $\operatorname{conv}(\\{p_{2}\\}\cup
S^{\prime})$ is a $k$-face of $P$. Thus, we can apply Lemma 1, and conclude
that if $P$ has minimal $k$-content, then it is symmetric to the bisector of
$[p_{1},p_{2}]$. Applying the same argument for all edges of
$\operatorname{conv}(S_{1}\cup Q)$ we obtain that
$\operatorname{conv}(S_{1}\cup Q)$ is a regular simplex and the affine hull of
$S_{2}$ meets it at its center, showing that in this case $Q=\emptyset$.

Using the same argument, we obtain that for any values of $k_{1}$ and $k_{2}$,
if $P$ has minimal total $k$-content, then it is symmetric to the bisector of
every edge of $S_{1}$, $S_{2}$, and if it exists, of $Q$. This yields the
assertion. ∎

By the idea in the proof of Theorem 7, we obtain Theorem 8.

######  Theorem 8.

For any $1\leq k\leq d-1$, among simplices in $\mathbb{E}^{d}$ with unit
inradius, the regular ones have minimal total $k$-content. Furthermore, for
any $1\leq k\leq d-1$, among the convex polytopes in $\mathbb{E}^{d}$ with
$d+2$ vertices and unit inradius, if $P$ has minimal total $k$-content, then
it satisfies one of the following.

  * (i)

There are regular simplices $S_{1}$ and $S_{2}$ with $\dim S_{1}+\dim S_{2}=d$
such that the subspaces spanned by $S_{1}$ and $S_{2}$ are orthogonal
complements of each other, $S_{1}\cap S_{2}=\\{c\\}$, and
$P=\operatorname{conv}(S_{1}\cup S_{2})$.

  * (ii)

There are regular simplices $S_{1},S_{2}$, of dimensions $\dim S_{1}=k_{1}\leq
k$ and $\dim S_{2}=k_{2}\leq k$ and lying in orthogonal subspaces that meet at
the common center $q_{1}$ of both simplices, and a regular
$(d-k_{1}-k_{2})$-dimensional simplex $Q$ with center $q_{2}$ such that $Q$ is
perpendicular to the segment $[q_{1},q_{2}]$, and
$P=\operatorname{conv}(S_{1}\cup S_{2}\cup Q)$.

  * (iii)

$P$ is a regular simplex of unit volume.

##  4\. Applications for measures with rotationally symmetric density
functions

Theorems 2 and 3 are immediate consequences of the following, more general
theorems, which can be proved by the same idea (for the proof of Theorem 9,
see also the proof of [2, Lemma 9]).

######  Theorem 9.

Let $D$ be an open ball centered at $o$, or let $D=\mathbb{E}^{d}$. Let
$\rho:[0,R_{0})$ be a nonnegative function, where
$R_{0}\in\mathbb{R}\cup\\{\infty\\}$ denotes the radius of $D$. Let
$S_{\mathrm{reg}}\subset D$ be a regular $d$-dimensional simplex centered at
$o$.

  * (i)

If $\rho$ is a decreasing function, then for any simplex $S\subset D$ with
$\operatorname{cr}(S)=\operatorname{cr}(S_{\mathrm{reg}})$, we have

$\int_{S}\rho(||x||)\,dx\geq\int_{S_{\mathrm{reg}}}\rho(||x||)\,dx.$

Furthermore, if $\rho$ is strictly decreasing, then equality holds if and only
if $S$ is congruent to $S_{\mathrm{reg}}$, and its center is $o$.

  * (ii)

If $\rho$ is increasing, then for any $d$-dimensional simplex $S\subset D$
with $\operatorname{ir}(S)=\operatorname{ir}(S_{\mathrm{reg}})$, we have

$\int_{S}\rho(||x||)\,dx\leq\int_{S_{\mathrm{reg}}}\rho(||x||)\,dx.$

Furthermore, if $\rho$ is strictly increasing, then equality holds if and only
if $S$ is congruent to $S_{\mathrm{reg}}$, and its center is $o$.

######  Theorem 10.

Let $D$ be an open ball centered at $o$, or let $D=\mathbb{E}^{d}$. Let
$\rho:[0,R_{0})$ be a nonnegative function, where
$R_{0}\in\mathbb{R}\cup\\{\infty\\}$ denotes the radius of $D$. Let $P\subset
D$ be a convex polytope with $d+2$ vertices.

  * (i)

If $\rho$ is a decreasing function, then there is a convex polytope
$Q=\operatorname{conv}(S_{1}\cup S_{2})$ of circumradius equal to
$\operatorname{cr}(P)$, where $S_{1}$ and $S_{2}$ are regular simplices
centered at $o$ such that the subspaces spanned by $S_{1}$ and $S_{2}$ are
orthogonal complements of each other, and

$\int_{P}\rho(||x||)\,dx\geq\int_{Q}\rho(||x||)\,dx.$

  * (ii)

If $\rho$ is a decreasing function, then there is a convex polytope
$Q=\operatorname{conv}(S_{1}\cup S_{2})$ of inradius equal to
$\operatorname{ir}(P)$, where $S_{1}$ and $S_{2}$ are regular simplices
centered at $o$ such that the subspaces spanned by $S_{1}$ and $S_{2}$ are
orthogonal complements of each other, and

$\int_{P}\rho(||x||)\,dx\leq\int_{Q}\rho(||x||)\,dx.$

We remark that since by central projection, an open hemisphere in
$\mathbb{S}^{d}$ can be represented as the Euclidean space $\mathbb{E}^{d}$
equipped with a strictly decreasing rotationally symmetric density function,
Theorems 2 and 3 can be also applied for spherical polytopes. As an example
for another application, we present Corollary 2.

######  Corollary 2.

Recall that the density function of the standard Gaussian measure on
$\mathbb{E}^{d}$ is $\rho(||x||)$ with
$\rho(\tau)=\frac{1}{(2\pi)^{d/2}}e^{-\tau^{2}/2}$. Since $\rho(\tau)$ is a
strictly decreasing function of $\tau$, by Theorem 9 we have that among
simplices in $\mathbb{E}^{d}$ with a given circumradius, the regular simplex
centered at the origin has maximal Gaussian measure. By Theorem 10, we obtain
a similar statement for the Gaussian measures of polytopes in $\mathbb{E}^{d}$
with $d+2$ vertices and a given circumradius.

##  5\. Concluding remarks and open problems

Motivated by Theorem 9, it is a natural question to ask if for any measure
generated by a rotationally symmetric density function $\rho$, if $B$ is a
ball centered at $o$, then among simplices $S$ contained in $B$,
$\int_{S}\rho(||x||)dx$ is maximal if $S$ is a regular simplex inscribed in
$B$. Our next example shows that this is not true in general.

######  Example 1.

Let $B$ be the closed unit disk centered at $o$, let $0<\varepsilon\leq 1$,
and set

$\rho(\tau)=\left\\{\begin{array}[]{l}1,\hbox{ if
}\tau\in[1-\varepsilon,1],\\\ 0,\hbox{ otherwise.}\end{array}\right.$

Let $T_{\mathrm{reg}}$ denote a regular triangle inscribed in $B$, and let $T$
denote an isosceles triangle inscribed in $B$ such that the length of its
altitude belonging to its base is $\varepsilon$. An elementary computation
shows that $\int_{T_{\mathrm{reg}}}\rho(||x||)dx=\Theta(\varepsilon^{2})$, and
$\int_{T}\rho(||x||)dx=\Theta(\varepsilon^{3/2})$. Thus, if $\varepsilon$ is
sufficiently small,
$\int_{T_{\mathrm{reg}}}\rho(||x||)dx<\int_{T}\rho(||x||)dx$.

We could not prove that among simplices in $\mathbb{S}^{d}$ containing a given
ball $B$, the regular ones circumscribed about $B$ have minimal volume.
Nevertheless, we show that an affirmative answer to this problem implies a
spherical variant of the so-called Simplex Mean Width Conjecture (see e.g.
[17]), stating that among simplices contained in a given ball $B$ in
$\mathbb{E}^{d}$, the regular ones inscribed in $B$ have maximal mean width.
Before introducing it, we remark that the mean width of a convex body is
proportional to its first intrinsic volume, and note that, apart from the
equality case, by the property described in Remark 6 an affirmative answer to
Problem 1 implies the Simplex Mean Width Conjecture by a standard limit
argument.

######  Remark 6.

A detailed description of possible variants of intrinsic volumes of convex
bodies in $\mathbb{S}^{d}$ can be found in [11]. One of these, based on the
geometric observation that the first intrinsic volume of a convex body in
$\mathbb{E}^{d}$ is proportional to the total rotation invariant measure of
the hyperplanes intersecting it, is defined as follows. Let $\mathcal{S}$
denote the space of spherical hyperplanes in $\mathbb{S}^{d}$, let $\mu$
denote the rotation invariant probability measure on $\mathcal{S}$, and let
$\chi(\cdot)$ define Euler characteristic. For any convex body in
$\mathbb{S}^{d}$, set

$U_{1}(K)=\frac{1}{2}\int_{\mathcal{S}}\chi(K\cap S)\,d\nu(S).$

By (20) of [11], for any spherically convex set $K\subset\mathbb{S}^{d}$, we
have

$U_{1}(K)=\frac{1}{2}-\frac{\operatorname{vol}_{d}(K^{*})}{\operatorname{vol}_{d}(\mathbb{S}^{d})}$

where $K^{*}$ denotes the polar of $K$. Thus, in the family of simplices in
$\mathbb{S}^{d}$ contained in a given ball $B\subset\mathbb{S}^{d}$, the
functional $U_{1}$ attains its maximum at $S$ if and only if among the
simplices containing $B^{*}$, the simplex $S^{*}$ has minimal volume.

######  Problem 1.

Let $d\geq 4$. Prove or disprove that for any ball $B\subset\mathbb{S}^{d}$,
among the simplices containing $B$ the regular ones circumscribed about $B$
have minimal volume.

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