# Steiner symmetrization on the sphere

Bushra Basit  ,  Steven Hoehner  ,  Zsolt Lángi  and  Jeff Ledford  Bushra
Basit, Department of Algebra and Geometry, Budapest University of Technology
and Economics, Műegyetem rkp. 3, H-1111 Budapest, Hungary
<EMAIL_ADDRESS>Steven Hoehner, Department of Mathematics & Computer
Science, Longwood University, 201 High Street, Farmville, VA 23909, U.S.A.
<EMAIL_ADDRESS>Zsolt Lángi, Department of Algebra and Geometry,
Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1111
Budapest, Hungary, and HUN-REN Alfréd Rényi Institute of Mathematics,
Reáltanoda utca 13-15, H-1053, Budapest, Hungary<EMAIL_ADDRESS>Jeff
Ledford, Department of Mathematics & Computer Science, Longwood University,
201 High Street, Farmville, VA 23909, U.S.A<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to introduce a generalization of Steiner
symmetrization in Euclidean space for spherical space, which is the dual of
the Steiner symmetrization in hyperbolic space introduced by Peyerimhoff (J.
London Math. Soc. (2) 66: 753-768, 2002). We show that this symmetrization
preserves volume in every dimension, and convexity in the spherical plane, but
not in dimensions $n>2$. In addition, we investigate the monotonicity
properties of the perimeter and diameter of a set under this process, and find
conditions under which the image of a spherically convex disk under a suitable
sequence of Steiner symmetrizations converges to a spherical cap. We apply our
results to prove a spherical analogue of a theorem of Sas, and to confirm a
conjecture of Besau and Werner (Adv. Math. 301: 867-901, 2016) for centrally
symmetric spherically convex disks.

###### Key words and phrases:

Spherical convex body, Steiner symmetrization, inscribed polytope, theorem of
Macbeath, theorem of Sas, floating area

###### 2020 Mathematics Subject Classification:

52A55 (52A40, 28A75, 53A35)

Partially supported by the ERC Advanced Grant “ERMiD”, and the NKFIH grant
K147544.

##  1\. Introduction

Steiner symmetrization of convex bodies in the $n$-dimensional Euclidean space
$\mathbb{R}^{n}$ is a powerful tool to solve geometric optimization problems.
It has many desirable properties which lend to its usefulness. In particular,
it preserves volume and is nonincreasing for the other quermassintegrals as
well as the diameter, the symmetrization of a convex body is a convex body,
and for any convex body there exists a sequence of Steiner symmetrizations
that converges to a Euclidean ball. These properties allow one to use the
Steiner symmetrization process to prove many fundamental results in convex
geometry, including the isoperimetric, isodiametric and Brunn-Minkowski
inequalities. For background on the geometric aspects of Steiner
symmetrizations of convex bodies, we refer the reader to, for example, the
books of Gruber [12] and Schneider [21].

Due to its usefulness, there have been attempts to generalize the concept of
Steiner symmetrization to other spaces of constant curvature. In particular,
Peyerimhoff [19] introduced a variant of Steiner symmetrization in the
hyperbolic space $\mathbb{H}^{n}$ which preserves volume, but the Steiner
symmetral of a convex body is in general not convex. Furthermore, Böröczky [5]
defined Steiner symmetrization in the spherical space $\mathbb{S}^{n}$, which
increases volume and does not preserve convexity (For more information on
these symmetrizations, see [1]).

In this paper we introduce a new variant of Steiner symmetrization on
$\mathbb{S}^{n}$, which is the ‘conceptual dual’ of the one introduced by
Peyerimhoff [19]. We show that it preserves volume, and on
$\operatorname{\mathbb{S}}^{2}$, on the family of spherically convex disks,
Steiner symmetrizations satisfying certain additional properties preserve
convexity. We investigate the properties of this symmetrization process, and
as an application, we prove a theorem of Sas [20] about the area of polygons
inscribed in a convex disk. In addition, we prove a conjecture of Besau and
Werner [3] about the floating area of convex bodies in $\mathbb{S}^{n}$, for
centrally symmetric convex disks.

The structure of the paper is as follows. In Section 2, we introduce our new
symmetrization process, and investigate its properties. More specifically, in
Subsection 2.1 we give a brief overview of the properties of spherical
geometry that we are going to use in the paper, and define our symmetrization
on $\mathbb{S}^{n}$. In Subsection 2.2 we show that spherical volume does not
change under symmetrization in $\mathbb{S}^{n}$. In Subsection 2.3, we prove
that if $K$ is a spherically convex disk $K$ in
$\operatorname{\mathbb{S}}^{2}$ satisfying the so-called _angular monotonicity
property_ with respect to the symmetrization, then its image is spherically
convex. We also show that this result does not hold in $\mathbb{S}^{n}$ for
any $n>2$. In Subsections 2.4 and 2.5, we show that the perimeter and the
diameter of a convex disk in $\operatorname{\mathbb{S}}^{2}$ does not increase
under Steiner symmetrization. Finally, in Subsection 2.6 we prove that if $K$
is a centrally symmetric convex disk in $\operatorname{\mathbb{S}}^{2}$, or
its diameter is less than $\frac{\pi}{2}$, then applying subsequent Steiner
symmetrizations on $K$ a spherical cap of the same area can be approximated
arbitrarily well in the Hausdorff metric.

In Section 3 we deal with applications of our symmetrization process. In
particular, in Subsection 3.1 we prove that for any $N\geq 3$ and $0<A<2\pi$,
among centrally symmetric convex disks $K$ of area $A$, the area of a maximum
area convex $N$-gon contained in $K$ is minimal if $K$ is a spherical cap.
This result is a generalization of a theorem of Sas [20], proving the same
property of Euclidean circular disks in $\mathbb{R}^{2}$ in the family of
convex disks. In Subsection 3.2, we finish the paper by verifying a conjecture
of Besau and Werner [3] for centrally symmetric convex disks, about an
isoperimetric property of spherical balls regarding their floating areas.

##  2\. Steiner symmetrization on the sphere

###  2.1. Preliminaries

This paper investigates problems in the $n$-dimensional spherical space
$\mathbb{S}^{n}$, which we regard as the unit sphere in the
$(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$, centered at the origin
$o$. More specifically, if $\langle\cdot,\cdot\rangle$ denotes the standard
inner product of $\mathbb{R}^{n+1}$ and $\|x\|=\sqrt{\langle x,x\rangle}$
denotes the Euclidean norm of $x\in\mathbb{R}^{n+1}$, then we set
$\mathbb{S}^{n}=\\{x\in\mathbb{R}^{n+1}:\|x\|=1\\}$. The _spherical distance_
of points $x,y\in\mathbb{S}^{n}$ is defined as

$d_{s}(x,y)=\arccos\left(\langle x,y\rangle\right).$

Here and throughout the paper, unless we state otherwise, by “distance” we
always mean “spherical distance”. Points of $\mathbb{S}^{n}$ at distance $\pi$
are called _antipodal_. A pair of points is antipodal if it is the
intersection of $\mathbb{S}^{n}$ and a line in $\mathbb{R}^{n+1}$ through $o$.
More generally, the intersection of $\mathbb{S}^{n}$ with a
$(k+1)$-dimensional linear subspace of $\mathbb{R}^{n+1}$, where $k\geq 0$, is
called a _$k$ -dimensional subspace_ or _greatsphere_ of $\mathbb{S}^{n}$. We
call $1$-dimensional greatspheres _greatcircles_ , or _(spherical) lines_. For
any distinct points $x,y\in\operatorname{\mathbb{S}}^{n}$, there is a unique
line containing them. If $x,y$ are not antipodal, then they decompose this
line into two closed noncongruent arcs; the shorter such arc, denoted by
$[x,y]_{s}$, is called the _segment_ with endpoints $x,y$. For any $0<r<\pi$
and $x\in\mathbb{S}^{n}$, the set
$B_{s}(x,r)=\\{y\in\mathbb{S}^{n}:\,d_{s}(x,y)\leq r\\}$ is called the _closed
(spherical) ball_ of radius $r$ and center $x$. The (relative) interior of a
closed ball is called an _open ball_. Closed/open balls with $r=\pi/2$ are
called closed/open _hemispheres_. The (relative) boundary of a set
$S\subset\mathbb{S}^{n}$ is denoted $\operatorname{bd}(S)$.

A set $K\subset\mathbb{S}^{n}$ is called _(spherically) convex_ if it is
contained in an open hemisphere, and for every $x,y\in K$, the segment
$[x,y]_{s}$ is contained in $K$. The intersection of convex sets in
$\mathbb{S}^{n}$ is convex. In particular, for any set
$A\subset\mathbb{S}^{n}$ contained in an open hemisphere, the intersection of
all convex sets containing $A$ is convex; this set is called the _(spherical)
convex hull_ of $A$, denoted by $\operatorname{conv}_{s}(A)$. If
$A\subset\mathbb{S}^{n}$ is a finite set contained in an open hemisphere, the
set $P=\operatorname{conv}_{s}(A)$ is called a _(spherical) polytope_. In this
case, if $A$ satisfies the condition that $\operatorname{conv}_{s}(B)\neq P$
for any $B\subsetneq A$, then $A$ is called a _minimal representation_ of $P$.
Every polytope has a unique minimal representation, and its elements are
called the _vertices_ of $P$.

A compact, spherically convex set with nonempty interior is called a
_(spherical) convex body_. The $n$-dimensional Hausdorff measure of a
spherical convex body $K$ is called its _(spherical) volume_ , denoted by
$\operatorname{vol}_{s}(K)$. In the case $n=2$, we write
$\operatorname{area}(K)$ for $\operatorname{vol}_{s}(K)$.

In the following we fix some open hemisphere $S$ of $\mathbb{S}^{n}$ with
center $c$, and denote the family of all convex bodies of $\mathbb{S}^{n}$
contained in $S$ by $\mathcal{K}_{S}$. Let $\bar{L}$ and $\bar{H}$ be a
greatcircle and an $(n-1)$-dimensional greatsphere, respectively, meeting
orthogonally at $c$, and set $L=\bar{L}\cap S$, $H=\bar{H}\cap S$. For any
$2$-dimensional greatsphere $\bar{G}$ through $\bar{L}$ and
$0<\delta<\frac{\pi}{2}$, the points of $G$ at spherical distance $\delta>0$
from $\bar{L}$ consists of two circles $\bar{C}_{1},\bar{C}_{2}$ of spherical
radius $\frac{\pi}{2}-\delta$, each concentric with one of the closed
hemispheres in $\bar{G}$ bounded by $\bar{L}$. We call the curves
$C_{i}=\bar{C}_{i}\cap S$, $i=1,2$, _distance curves_ with axis $L$ and
distance $\delta$. We regard $L$ as the only distance curve with axis $L$ and
distance $0$, and remark that any point $x\in S$ belongs to a unique distance
curve with axis $L$. For later use, we denote the set of points of $\bar{H}$
at spherical distance $\frac{\pi}{2}$ from $L$ by $H_{0}$; this set is an
$(n-2)$-dimensional greatsphere of $\mathbb{S}^{n}$, contained in
$\operatorname{bd}(S)$.

Our main definition is the following.

######  Definition 2.1.

Let $X\subset S$ be compact. Assume that for any distance curve $C$ with axis
$L$, the set $C\cap X$ is connected. Then the _Steiner symmetral_ of $X$ with
respect to $(L,H)$, denoted by $\sigma_{L,H}(X)$, is defined as follows. For
any distance curve $C$ with axis $L$:

  1. (i)

if $X\cap C=\emptyset$, then $\sigma_{L,H}(X)\cap C=\emptyset$;

  2. (ii)

if $X\cap C$ is a singleton, then $\sigma_{L,H}(X)\cap C=C\cap H$;

  3. (iii)

if $X\cap C$ is a closed circle arc, then $\sigma_{L,H}(X)\cap C$ is the
unique closed circle arc in $S$ with the same length, centered at $C\cap H$.

Motivated by Definition 2.1, from now on, if a set $X\subset S$ satisfies the
property that for any distance curve $C$ with axis $L$ the set $X\cap C$ is
connected, then we say that $X$ satisfies the _connectedness property_.

######  Remark 2.2.

By construction, $\sigma_{L,H}(K)$ is symmetric with respect to $H$ and is
invariant on $H$-symmetric sets in $S$. It is also monotonic, i.e., if the
compact sets $K\subseteq L$ satisfy the connectedness property, then
$\sigma_{L,H}(K)\subset\sigma_{L,H}(L)$. Furthermore, Steiner symmetrization
is idempotent, meaning that $\sigma_{L,H}(\sigma_{L,H}(K))=\sigma_{L,H}(K)$
holds for any compact set $K$ with the connectedness property.

######  Remark 2.3.

We note that any spherical ball satisfies the connectedness property. On the
other hand, this is not true in general for every spherically convex body in
$S$. To provide such an example for $n=2$, we may take a spherical polygon not
intersecting $L$ which has more than one vertex closest to $L$. For $n>2$, a
straightforward modification of this example shows this observation.

######  Remark 2.4.

An elementary observation shows that if $K\subset S$ is a convex body with
$c\in K$, or more generally, if $K\cap L\neq\emptyset$, then $K$ satisfies the
connectedness property. Indeed, if $q_{1},q_{2}\in K\cap C$, then by
convexity, the spherical triangle with vertices $q_{1},q_{2},c$ contains the
arc of $C$ between $q_{1}$ and $q_{2}$, and thus the set $C\cap K$ is
connected.

######  Definition 2.5.

For any set $X\subset S$, the _projection of $X$ onto $H$ along $L$_ is the
set of points $q\in H$ with the property that the distance curve through $q$
with axis $L$ intersects $X$. We denote this set by
$\operatorname{proj}_{L,H}(X)$.

In all parts of the paper, apart from Subsection 2.2 and Subsubsection 2.3.2,
we deal with the spherical plane $\operatorname{\mathbb{S}}^{2}$. In all of
these parts, we use the following coordinate system on $\mathbb{R}^{3}$:
$c=(1,0,0)$, $L$ lies in the $(x,y)$-plane, and $H$ lies in the $(x,z)$-plane.
We use the normal polar coordinates for any point $p\in S$, namely
$\theta_{p}$ denotes the (signed) spherical distance of $p$ from $L$, and
$\varphi_{p}$ denotes the (signed) spherical distance of the orthogonal
projection of $p$ onto $L$ from $H$, implying also that
$-\frac{\pi}{2}<\theta_{p},\varphi_{p}<\frac{\pi}{2}$. We call the points
$(0,0,1)$ and $(0,0,-1)$ the _poles_ of $\operatorname{\mathbb{S}}^{2}$, and
for any $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$, we denote the distance curve
$\\{p:\theta_{p}=\theta\\}$ by $C_{\theta}$.

###  2.2. Volume of the Steiner symmetral of a set

######  Theorem 2.6.

For any compact set $X\subset S$ with the connectedness property,
$\operatorname{vol}_{s}(X)=\operatorname{vol}_{s}(\sigma_{L,H}(X))$.

###### Proof.

Let $L_{1}$ denote the linear hull of $\bar{L}$, and let $L_{2}$ be the
orthogonal complement of $L_{1}$ in $\mathbb{R}^{n+1}$. Note that then
$H_{0}=L_{2}\cap\mathbb{S}^{n}$ is the set of points of $\bar{H}$ at spherical
distance $\frac{\pi}{2}$ from $\bar{L}$, and thus, it is a subset of
$\operatorname{bd}(S)$. Clearly, any point $p\in\mathbb{R}^{n+1}$ can be
uniquely written as $p=p_{1}+p_{2}$ with $p_{i}\in L_{i}$ for $i=1,2$,
implying that any point $q\in\mathbb{S}^{n}$ can be written as
$q=(\cos\theta)q_{1}+(\sin\theta)q_{2}$, where $q_{1}\in\bar{L}$, $q_{2}\in
H_{0}$, and $0\leq\theta\leq\frac{\pi}{2}$. Furthermore, this representation
is unique if $\theta\neq 0,\frac{\pi}{2}$, and the property that $q\in S$ is
equivalent to the condition that $q_{1}\in L$.

In the following, we use a Descartes coordinate system of $\mathbb{R}^{n+1}$
in which the the plane spanned by the first two basis vectors is $L_{1}$, and
the $(n-1)$-dimensional linear subspace spanned by the remaining basis vectors
is $L_{2}$. We write each point $x=(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}$
as $x=(r\cos\theta\cos\phi,r\cos\theta\sin\phi,(r\sin\theta)q_{2})$, where
$r\geq 0$, $0\leq\theta\leq\frac{\pi}{2}$, $-\pi\leq\phi\leq\pi$, and
$q_{2}=(q_{2,1},\ldots,q_{2,n-1})$. The Jacobian determinant $\det(J)$ of this
coordinate transformation on $\mathbb{R}^{n+1}$ can be expressed as

$\displaystyle\det(J)$
$\displaystyle=\det\left(\frac{\partial(x_{1},\ldots,x_{n+1})}{\partial(r,\theta,\phi,q_{2,1},\ldots,q_{2,n-1})}\right)$
$\displaystyle=\det\left(\begin{array}[]{ccc|ccc}\cos\theta\cos\phi&-r\sin\theta\cos\phi&-r\cos\theta\sin\phi&0&\cdots&0\\\
\cos\theta\sin\phi&-r\sin\theta\sin\phi&r\cos\theta\cos\phi&0&\cdots&0\\\
\hline\cr(\sin\theta)q_{2,1}&(r\cos\theta)q_{2,1}&0&&&\\\
\vdots&\vdots&\vdots&&(r\sin\theta)D(q_{2})&\\\
(\sin\theta)q_{2,n-2}&(r\cos\theta)q_{2,n-2}&0&&&\end{array}\right),$

where $D(q_{2})$ is the $(n-1)\times(n-2)$ derivative matrix of $q_{2}$. From
this we obtain

$\displaystyle\det(J)$
$\displaystyle=(r^{n}\sin^{n-2}\theta)\det\left(\begin{array}[]{ccc|ccc}\cos\theta\cos\phi&-\sin\theta\cos\phi&-\cos\theta\sin\phi&0&\cdots&0\\\
\cos\theta\sin\phi&-\sin\theta\sin\phi&\cos\theta\cos\phi&0&\cdots&0\\\
\hline\cr(\sin\theta)q_{2,1}&(\cos\theta)q_{2,1}&0&&&\\\
\vdots&\vdots&\vdots&&D(q_{2})&\\\
(\sin\theta)q_{2,n-1}&(\cos\theta)q_{2,n-1}&0&&&\end{array}\right)$
$\displaystyle=-(r^{n}\sin^{n-2}\theta\cos\theta\sin\phi)\det\left(\begin{array}[]{cc|ccc}\cos\theta\sin\phi&-\sin\theta\sin\phi&0&\cdots&0\\\
\hline\cr(\sin\theta)q_{2,1}&(\cos\theta)q_{2,1}&&&\\\
\vdots&\vdots&&D(q_{2})&\\\
(\sin\theta)q_{2,n-1}&(\cos\theta)q_{2,n-1}&&&\end{array}\right)$
$\displaystyle-(r^{n}\sin^{n-2}\theta\cos\theta\cos\phi)\det\left(\begin{array}[]{cc|ccc}\cos\theta\cos\phi&-\sin\theta\cos\phi&0&\cdots&0\\\
\hline\cr(\sin\theta)q_{2,1}&(\cos\theta)q_{2,1}&&&\\\
\vdots&\vdots&&D(q_{2})&\\\
(\sin\theta)q_{2,n-2}&(\cos\theta)q_{2,n-2}&&&\end{array}\right)$
$\displaystyle=-(r^{n}\sin^{n-2}\theta\cos\theta\sin\phi)\bigg{(}(\cos\theta\sin\phi)\det\left((\cos\theta)q_{2}^{T};D(q_{2})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+(\sin\theta\sin\phi)\det\left((\sin\theta)q_{2}^{T};D(q_{2})\right)\bigg{)}$
$\displaystyle\phantom{=}-(r^{n}\sin^{n-2}\theta\cos\theta\cos\phi)\bigg{(}(\cos\theta\cos\phi)\det\left((\cos\theta)q_{2}^{T};D(q_{2})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+(\sin\theta\cos\phi)\det\left((\sin\theta)q_{2}^{T};D(q_{2})\right)\bigg{)}$
$\displaystyle=-(r^{n}\sin^{n-2}\theta\cos\theta)\det\left(q_{2}^{T};D(q_{2})\right).$

Here we remark that $\det\left(q_{2}^{T};D(q_{2})\right)dq_{2}$ is the
(spherical) volume element of $H_{0}$, identified with $\mathbb{S}^{n-3}$.
Thus, by Fubini’s theorem, the spherical volume of $X\subset S$ is

(1) $\operatorname{vol}_{s}(X)=\int_{0}^{\pi/2}\int_{q_{2}\in H_{0}}l(X\cap
C(\theta,q_{2}))\,\sin^{n-2}\theta\cos\theta\,dq_{2}\,d\theta$

where $l(\cdot)$ denotes spherical arclength, and $C(\theta,q_{2})$ is the
distance curve through $q_{2}$ with axis $L$ and distance $\theta$; this curve
is the semicircle

$(\cos\theta\cos\varphi,\cos\theta\sin\varphi,(\sin\theta)q_{2}),\quad-\frac{\pi}{2}<\varphi<\frac{\pi}{2}.$

Note that the expression in (1) does not change under the Steiner
symmetrization $\sigma_{L,H}$, implying the assertion. ∎

###  2.3. Convexity of the Steiner symmetral

####  2.3.1. Convexity on $\operatorname{\mathbb{S}}^{2}$

Recall that in this part we use the coordinate system introduced in the last
paragraph of Subsection 2.1.

######  Definition 2.7.

Let $C\subset S$ be an arc of a distance curve with axis $L$. Let $p,q$ denote
the endpoints of $C$. Then the quantity $|\varphi_{p}-\varphi_{q}|$ is called
the _angular length_ of $C$.

######  Definition 2.8.

Let $K\in\mathcal{K}_{S}$. Assume that the angular length of $K\cap
C_{\theta}$ is nonincreasing for $\theta\geq 0$, and nondecreasing for
$\theta\leq 0$. Then we say that $K$ satisfies the _angular monotonicity
property_.

######  Lemma 2.9.

Let $K\in\mathcal{K}_{S}$. Then $K$ satisfies the angular monotonicity
property if and only if $K$ intersects $L$ in a spherical segment
$[q_{1},q_{2}]_{s}$, and $K$ has a pair of supporting lines $L_{1},L_{2}$ at
$q_{1},q_{2}$, respectively, such that $L_{1},L_{2}$ are symmetric to the
midpoint of $[q_{1},q_{2}]_{s}$.

By a _lune_ , we mean the intersection of two closed hemispheres of
$\mathbb{S}^{n}$, bounded by different greatspheres. Before proving Lemma 2.9,
we make the following observation.

######  Remark 2.10.

The condition of the lemma for $L_{1},L_{2}$ is equivalent to the property
that the midpoint of $[q_{1},q_{2}]_{s}$ is the center of the lune in $S$
bounded by $L_{1}$ and $L_{2}$, and also to the property that if $z$ is any of
the two intersection points of $L_{1}$ and $L_{2}$, then the sum of the angles
of the triangle $\operatorname{conv}_{s}\\{q_{1},q_{2},z\\}$ at $q_{1}$ and
$q_{2}$ is equal to $\pi$.

###### Proof of Lemma 2.9.

We only prove the ‘if’ part, as the opposite direction can be shown by a
straightforward modification of our argument.

Assume that $K$ intersects $L$ in a spherical segment $[q_{1},q_{2}]_{s}$, and
that it has a pair of supporting lines $L_{1},L_{2}$ satisfying the condition
in the lemma. Let $h$ denote the rotation around the poles satisfying
$h(q_{1})=q_{2}$. Then by our conditions, $h(L_{1})=L_{2}$. This shows that
the angular length of an arc $C\cap K$ is the same for any distance curve $C$
with axis $L$. In particular, this implies Lemma 2.9 for the special case that
$K$ is a lune.

Now let $q_{1}^{\prime},q_{2}\in[q_{1},q_{2}]_{s}$ be such that
$q_{1}^{\prime}$ is closer to $q_{1}$ than $q_{2}^{\prime}$. Let
$L_{1}^{\prime}$ and $L_{2}^{\prime}$ be the rotated copies of $L_{1}$ by the
rotation around the poles that maps $q_{1}$ into $q_{1}^{\prime}$ and
$q_{2}^{\prime}$, respectively. By the definition of spherical convexity, for
$i=1,2$, $L_{i}^{\prime}$ decomposes $\operatorname{bd}(K)$ into two connected
arcs $\Gamma_{1}$ and $\Gamma_{2}$, exactly one of them containing $q_{1}$. We
denote this arc by $\Gamma_{1}$. Clearly, by our conditions,
$\Gamma_{1}\subseteq\Gamma_{2}$.

On the other hand, the set of points $p\in\operatorname{bd}(K)$ with
$\theta_{p}\geq\theta$ is also a connected arc of $\operatorname{bd}(K)$,
which we denote by $\Psi_{\theta}$. Observe that the endpoints of
$\Psi_{\theta}$ are the endpoints of the arc $C_{\theta}\cap K$. This shows
that if $\theta_{2}>\theta_{1}$, then
$\Psi_{\theta_{2}}\subseteq\Psi_{\theta_{1}}$. Combining this with the
observation in the previous paragraph, we obtain that if
$\theta_{2}>\theta_{1}\geq 0$, then the angular length of $C_{\theta_{1}}\cap
K$ is not less than that of $C_{\theta_{2}}\cap K$. If
$0\geq\theta_{2}>\theta_{1}$, a similar observation yields the assertion. ∎

######  Remark 2.11.

By Lemma 2.9, if $K\in\mathcal{K}_{S}$ is symmetric about $c$, then $K$
clearly satisfies the angular monotonicity property.

######  Theorem 2.12.

Assume that $K\in\mathcal{K}_{S}$ satisfies the angular monotonicity property.
Then $\sigma_{L,H}(K)$ is convex.

###### Proof.

In the proof, for any $\theta_{1}<\theta_{2}$, we call the connected region of
$S$ bounded by the distance curves $C_{\theta_{1}}$ and $C_{\theta_{2}}$ a
_horizontal strip_ of $S$. Note that by a standard continuity argument, it is
sufficient to prove the statement for the case that $K$ is a convex polygon.

Suppose for contradiction that there is a polygon $P$ such that
$\sigma_{L,H}(P)$ is not convex. For any distance curve $C_{\theta}$, let us
denote the endpoints of $C_{\theta}\cap\sigma_{L,H}(P)$ by $p^{-}(\theta)$ and
$p^{+}(\theta)$ such that the $\varphi$-coordinate of $p^{-}(\theta)$ is less
than or equal to that of $p^{+}(\theta)$.

Note that if $\sigma_{L,H}(P)$ is not convex, then there is a horizontal strip
$T$, bounded by the distance curves $C_{\theta_{1}}$ and $C_{\theta_{2}}$,
where $\theta_{2}>\theta_{1}$ are nonnegative and are sufficiently close, such
that

  1. (i)

$T\cap\operatorname{bd}(P)$ consists of two segments, and

  2. (ii)

we have $\operatorname{area}(T\cap P)<\operatorname{area}(Q)$, where $Q$ is
the connected region bounded by $\sigma_{L,H}(C_{\theta_{1}}\cap P)$,
$\sigma_{L,H}(C_{\theta_{2}}\cap P)$, and the two segments
$[p^{-}(\theta_{1}),p^{-}(\theta_{2})]_{s}$ and
$[p^{+}(\theta_{1}),p^{+}(\theta_{2})]_{s}$.

For $i=1,2$, let $\varphi_{i}$ denote the angular length of
$C_{\theta_{i}}\cap P$. Then by the angular monotonicity property,
$\varphi_{2}\leq\varphi_{1}$. If $\varphi_{2}=\varphi_{1}$, then the segments
in $H\cap\operatorname{bd}(P)$ are rotated copies of each other by
$\varphi_{2}$ around the poles. Since Steiner symmetrization does not change
the angular length of $C_{\theta}\cap P$ for any value of $\theta$, in this
case $Q$ coincides with $\sigma_{L,H}(T\cap P)$, and Theorem 2.6 yields the
assertion.

Figure 1. An illustration for the dissection of $P$ in the proof of Theorem
2.12. The continuous line in the figure indicates $\operatorname{bd}(P)$. The
dashed segment in $P$, which is a rotated copy of the left segment in
$\operatorname{bd}(P)$ in the figure around the poles, decomposes $P$ into the
regions $P_{1},P_{2}$. The symbols $\varphi_{2},\varphi_{1}-\varphi_{2}$ show
the angular lengths of the corresponding distance curve arcs in the boundaries
of $P_{1}$ and $P_{2}$.

Assume that $\varphi_{2}<\varphi_{1}$. Then the rotated copy of one component
of $T\cap\operatorname{bd}(P)$ by $\varphi_{2}$ towards the other component
decomposes $T\cap P$ into two components $P_{1}$ and $P_{2}$ such that $P_{1}$
contains exactly one point of $C_{\theta_{2}}\cap P$ and an arc of angular
length $\varphi_{1}-\varphi_{2}$ of $C_{\theta_{1}}\cap P$, and $P_{2}$
contains two arcs of angular length $\varphi_{2}$ of $C_{\theta_{i}}\cap P$
for $i=1,2$ (see Figure 1). Note that we can obtain the Steiner symmetral of
$T\cap P$ by taking the Steiner symmetral of $P_{2}$, and rotating both curves
in its boundary, connecting the intersection point of $H$ and $C_{\theta_{2}}$
to a point of $C_{\theta_{1}}$, by $\frac{\varphi_{2}}{2}$ around the poles
and away from $H$. In other words, we may assume that $\varphi_{2}=0$,
implying that $p^{-}(\theta_{2})=p^{-}(\theta_{2})$ is the intersection point
of $H$ and $C_{\theta_{2}}$. Let us denote this common point by
$p(\theta_{2})$. Now, recall from spherical geometry that for any fixed
segment $[x,y]_{s}$, the points $z$ with the property that the area of the
triangle $\operatorname{conv}_{s}\\{x,y,z\\}$ is fixed, moves on a circle
containing $-x,-y$. This circle is called a _Lexell circle_ of $[x,y]_{s}$.
Observe that the Lexell circle of $[p^{-}(\theta_{1}),p^{+}(\theta_{1})]_{s}$
intersecting $C_{\theta_{2}}$, belonging to smallest area triangles, passes
through $p_{\theta_{2}}$. This contradicts the assumption that
$\operatorname{area}(T\cap P)<\operatorname{area}(Q)$. ∎

####  2.3.2. Convexity on $\mathbb{S}^{n}$ with $n>2$

In this subsubsection we investigate the convexity properties of the set
$\sigma_{L,H}(K)$ where $K$ is spherically convex. During this, we often use
the central projection of $S$ from $o$ onto the tangent hyperplane
$T_{c}\mathbb{S}^{n}$ of $\mathbb{S}^{n}$ at the center $c$ of $S$. We denote
this projection by $\pi_{c}:S\to T_{c}\mathbb{S}^{n}$. Recall that for any
$X\subseteq S$, $\pi_{c}(X)$ is convex if and only if $X$ is spherically
convex. To simplify our notation, for any $p\in S$, we set $p^{*}=\pi_{c}(p)$,
and we use the same notation for subsets of $S$.

######  Lemma 2.13.

Consider the distance curve $C_{\theta}$ of
$S\subset\operatorname{\mathbb{S}}^{2}$, where
$-\frac{\pi}{2}<\theta<\frac{\pi}{2}$. If $\theta=0$, i.e., if $C_{\theta}=L$,
then $\pi_{c}(C_{\theta})$ is the straight line

$z=0$

in the tangent plane $T_{c}\mathbb{S}^{2}=\\{x=1\\}$. If $\theta>0$, then
$\pi_{c}(C_{\theta})$ is the branch of the hyperbola

$1=\frac{z^{2}}{\tan^{2}\theta}-y^{2}.$

in $\\{x=1\\}$ with $z>0$. If $\theta<0$, then $\pi_{c}(C_{\theta})$ is the
branch of the same hyperbola with $z<0$.

###### Proof.

Let $q$ be a point of $C_{\theta}$, where we assume that $\theta>0$. Using
spherical polar coordinates, we have
$q=(\cos\theta\cos\varphi,\cos\theta\sin\varphi,\sin\theta)$ for some
$-\frac{\pi}{2}<\varphi<\frac{\pi}{2}$. Since the equation of
$T_{c}\operatorname{\mathbb{S}}^{2}$ is $\\{x=1\\}$, the central projection of
$q$ onto this plane is

$\pi_{c}(q)=\left(1,\tan\varphi,\frac{\tan\theta}{\cos\varphi}\right).$

Using the identity $1+\tan^{2}\alpha=\frac{1}{\cos^{2}\alpha}$, we may
eliminate $\varphi$ and obtain that the coordinates of $\pi_{c}(q)$ satisfy
the equations $x=1$ and $1=\frac{z^{2}}{\tan^{2}\theta}-y^{2}$. Since
$C_{\theta}$ is connected and $\pi_{c}$ is continuous, one can easily show
that $\pi_{c}(C_{\theta})$ is one of the two branches of the above hyperbola.
Now, since points with positive $\varphi$-coordinates on $S$ are projected
onto the open half-plane $\\{x=1\\}\cap\\{z>0\\}$, the assertion follows for
$\theta>0$. The remaining two cases can be proved by a similar argument. ∎

######  Lemma 2.14.

Let $T\subset S$ be any triangle with $c$ as a vertex. Then
$\operatorname{proj}_{L,H}(T)$ is convex.

###### Proof.

Note that $T\cup L$ is contained in a $3$-dimensional greatsphere. Since this
greatsphere contains every distance curve with axis $L$ through a point of
$T$, we may assume that it coincides with $\mathbb{S}^{n}$, i.e., that $n=3$.
For any point $q\in S$, we say that the _angular distance_ of $q$ from $H$ is
the angle of the rotation around $H_{0}$ that moves $q$ to $H$.

Furthermore, observe that if $T=\operatorname{conv}_{s}\\{c,q_{1},q_{2}\\}$
and $T^{\prime}=\operatorname{conv}_{s}\\{c,q_{1}^{\prime},_{2}^{\prime}\\}$,
where $[q_{1}^{\prime},q_{2}^{\prime}]_{s}$ is a rotated copy of
$[q_{1},q_{2}]_{s}$ around $H_{0}$, then
$\operatorname{proj}_{L,H}(T)=\operatorname{proj}_{L,H}(T^{\prime})$. Thus,
without loss of generality, we may assume that $H$ intersects
$[q_{1},q_{2}]_{s}$, and that $q_{1}$ and $q_{2}$ have the same angular
distance from $H$. For $i=1,2$, let $C_{i}$ denote the distance curve with
axis $L$ through $q_{i}$, and let
$\tilde{q}_{i}=\pi_{c}(\operatorname{proj}_{L,H}(q_{i}))$.

In the remaining part of the proof, we investigate the central projections of
$T$ and $\operatorname{proj}_{L,H}(T)$ onto
$T_{c}\operatorname{\mathbb{S}}^{3}$. We identify this tangent space with
$\mathbb{R}^{3}$, and, without loss of generality, assume that $\pi_{c}(L)$ is
the $z$-axis and $\pi_{c}(H)$ is the $(x,y)$-plane. Then, using a suitable
coordinate system, we have that
$\tilde{q}_{1}=(r_{1}\cos\varphi,r_{1}\sin\varphi,0)$ and
$\tilde{q}_{2}=(r_{2}\cos\varphi,-r_{2}\sin\varphi,0)$ for some
$r_{1},r_{2}>0$ and $0<\varphi<\frac{\pi}{2}$.

From Lemma 2.13, it follows that the points of the central projection
$C_{1}^{*}$ of the distance curve $C_{1}$ are of the form
$(r_{1}(z)\cos\varphi,r_{1}(z)\sin\varphi,z)$, where
$1=\frac{r_{1}(z)^{2}}{\tan^{2}\theta_{1}}-z^{2}$ and $\theta_{1}$ is the
spherical distance of $C_{1}$ from $L$. From this and the expressions for the
coordinates of $\tilde{q}_{1}$, we obtain that $r_{1}=\tan\theta_{1}$ and
$r_{1}(z)=\sqrt{1+z^{2}}r_{1}$, implying that
$q_{1}^{*}=(r_{1}\sqrt{1+\bar{z}^{2}}\cos\varphi,r_{1}\sqrt{1+\bar{z}^{2}}\sin\varphi,\bar{z})$
for some $\bar{z}\in\mathbb{R}$. Since the central projections of the points
of $S$ having a given angular distance from $H$ lie on two planes in
$\mathbb{R}^{3}$ parallel and symmetric to the $(x,y)$-plane, a similar
computation shows that then
$q_{2}^{*}=(r_{2}\sqrt{1+\bar{z}^{2}}\cos\varphi,-r_{2}\sqrt{1+\bar{z}^{2}}\sin\varphi,-\bar{z})$.
Without loss of generality, we may assume that $\bar{z}\geq 0$.

Let $m^{*}$ (resp. $\tilde{m}$) be the point where $[q_{1}^{*},q_{2}^{*}]$
(resp. $[\tilde{q}_{1},\tilde{q}_{2}]$) intersects the $(x,z)$-plane. Since
$\operatorname{proj}_{L,H}$ is a continuous function, it suffices to prove
that the spherical distance from $L$ of the distance curve through
$\pi_{c}^{-1}(m^{*})$ is not smaller than that of the curve through
$\pi_{c}^{-1}(\tilde{m})$.

We compute $\tilde{m}$. Clearly, $\tilde{m}=t\tilde{q}_{1}+(1-t)\tilde{q}_{2}$
for some $t\in(0,1)$, and the $y$-coordinate of $\tilde{m}$ is zero. From this
we obtain that $t=\frac{r_{2}}{r_{1}+r_{2}}$, implying that
$\tilde{m}=\left(\frac{2r_{1}r_{2}}{r_{1}+r_{2}}\cos\varphi,0,0\right)$.
Similarly,
$m^{*}=\left(\frac{2r_{1}r_{2}}{r_{1}+r_{2}}\sqrt{1+\bar{z}^{2}}\cos\varphi,0,\frac{\bar{z}(r_{2}-r_{1})}{r_{1}+r_{2}}\right)$.
Now, if $C_{\tilde{m}}$ denotes the central projection of the distance curve
with axis $L$ through $\pi_{c}^{-1}(\tilde{m})$, then the point $m^{\prime}$
of $C_{\tilde{m}}$ whose $z$-coordinate is equal to that of $m^{*}$ is
$m^{\prime}=\left(\frac{2r_{1}r_{2}}{r_{1}+r_{2}}\sqrt{1+z^{2}\left(\frac{r_{2}-r_{1}}{r_{1}+r_{2}}\right)^{2}}\cos\varphi,0,\frac{\bar{z}(r_{2}-r_{1})}{r_{1}+r_{2}}\right)$.
Thus, it suffices to show that

$0\leq\frac{2r_{1}r_{2}}{r_{1}+r_{2}}\sqrt{1+\bar{z}^{2}}\cos\varphi-\frac{2r_{1}r_{2}}{r_{1}+r_{2}}\sqrt{1+\bar{z}^{2}\left(\frac{r_{2}-r_{1}}{r_{1}+r_{2}}\right)^{2}}\cos\varphi.$

But this inequality readily follows from the inequality
$\left|\frac{r_{2}-r_{1}}{r_{1}+r_{2}}\right|<1$ for any $r_{1},r_{2}>0$. ∎

######  Remark 2.15.

Let $\Gamma\subset S\subset\operatorname{\mathbb{S}}^{3}$ be an arc of a
distance curve with axis $L$ and distance $\theta$. Let the signed angular
distances of the endpoints of $\Gamma$ from $H$ be $\varphi_{1},\varphi_{2}$;
here the angular distance of a point $q$ from $H$ is said to be positive if
$q$ belongs to a fixed connected component of $S\setminus H$, and it is said
to be negative if it is contained in the opposite component. We choose our
notation in such a way that $\varphi_{1}<\varphi_{2}$. Let
$\Gamma^{\prime}=\sigma_{L,H}(\Gamma)$, and note that the signed angular
distances of $\Gamma^{\prime}$ from $H$ are
$\pm\frac{\varphi_{2}-\varphi_{1}}{2}$.

Let $\tilde{\Gamma}$ and $\tilde{\Gamma}^{\prime}$ be the central projections
of $\Gamma$ and $\Gamma^{\prime}$ onto $\mathbb{R}^{3}$, respectively, where
we choose the coordinate system of $\mathbb{R}^{3}$ such that the projection
of $H$ is the $(x,y)$-plane, the projection of $L$ is the $z$-axis, and the
projections of points with positive angular distance from $H$ have positive
$z$-coordinate. By Lemma 2.13, if $z_{1}<z_{2}$ denote the $z$-coordinates of
the endpoints of $\tilde{\Gamma}$, we have $\varphi_{i}=\arctan z_{i}$ for
$i=1,2$. Thus, the $z$-coordinates of the endpoints of
$\tilde{\Gamma}^{\prime}$ are

$\pm\tan\frac{\arctan z_{2}-\arctan
z_{1}}{2}=\pm\frac{z_{2}-z_{1}}{\sqrt{1+z_{2}^{2}}\sqrt{1+z_{1}^{2}}+1+z_{1}z_{2}}.$

######  Theorem 2.16.

For any convex body $K\in\mathcal{K}_{S}$ with
$c\in\operatorname{proj}_{L,H}(K)$, $\operatorname{proj}_{L,H}(K)$ is convex.

###### Proof.

Note that if $c\in\operatorname{proj}_{L,H}(K)$, then
$\operatorname{proj}_{L,H}(K)=\operatorname{proj}_{L,H}(\operatorname{conv}_{s}(K\cup\\{c\\}))$,
and hence we can assume that $c\in K$. Consider any points
$q_{1},q_{2}\in\operatorname{proj}_{L,H}(K)$, and let $q\in[q_{1},q_{2}]_{s}$.
For $i=1,2$, choose some point $x_{i}\in K$ which belongs to
$\operatorname{proj}_{L,H}^{-1}(q_{i})$. Applying Lemma 2.14 for the triangle
$T=\operatorname{conv}_{s}\\{c,x_{1},x_{2}\\}$, we have
$q\in\operatorname{proj}_{L,H}(T)\subseteq\operatorname{proj}_{L,H}(K)$. ∎

We show that Theorem 2.12 does not hold in $\mathbb{S}^{n}$ for any $n>2$.

######  Theorem 2.17.

For any $n>2$, there is a $c$-symmetric convex body $K\in\mathcal{K}_{S}$ for
which $\sigma_{L,H}(K)$ is not convex.

###### Proof.

First, we construct such a body in $\operatorname{\mathbb{S}}^{3}$. Similar to
the proof of Lemma 2.14, we construct a set $K^{*}$ in $\mathbb{R}^{3}$ such
that its image $K=\pi_{c}^{-1}(K^{*})\subset S$ satisfies the required
conditions. With a little abuse of terminology, we identify any set
$X\subseteq S$ with its projection onto $\mathbb{R}^{3}$, or in other words,
we regard $\mathbb{R}^{3}$ as a model of the spherical space $S$. We use a
coordinate system in which $L$ is the $z$-axis and $H$ is the $(x,y)$-plane.
Furthermore, for any point $q=(u,v,0)$ of $H$, we denote by $C_{q}$ or
$C(u,v)$ the distance curve with axis $L$ passing through $q$. To work in this
model, we note that spherical planes passing through $H_{0}$ are represented
by planes parallel to the $(x,y)$-plane. We call these planes ‘horizontal’.
Thus, a rotation about $H_{0}$ moves a horizontal plane into another
horizontal plane, whereas it moves a non-horizontal plane into another non-
horizontal plane, which might not be parallel to the original one.

Consider a point $p_{0}=(x_{0},0,0)\in\mathbb{R}^{3}$ with $x_{0}>0$. Recall
(see Lemma 2.13 or the proof of Lemma 2.14) that the distance curve through
$p_{0}$ with axis $L$ is the set of points $(\sqrt{t^{2}+1}x_{0},0,t)$, where
$t$ runs over $\mathbb{R}$. Note also that in our model, intersections of
spherical planes with $S$ are represented by Euclidean planes. We choose two
planes $H_{+}$ and $H_{-}$ through $p_{0}$, with equations
$A_{+}(x-x_{0})+B_{+}y=z$ and $A_{-}(x-x_{0})+B_{-}y=z$, respectively. We call
$H_{+}$ and $H_{-}$ the _upward_ and _downward_ plane, respectively, and for
$*\in\\{+,-\\}$, label parameters related to $H_{*}$ with subscript $*$. We
choose specific values for the parameters in the equations later;
nevertheless, we intend to guarantee that $L$ is not disjoint from the
intersection of the open halfspaces
$\\{(x,y,z)\in\mathbb{R}^{3}:A_{+}(x-x_{0})+B_{+}y>z\\}$ and
$\\{(x,y,z)\in\mathbb{R}^{3}:A_{-}(x-x_{0})+B_{-}y<z\\}$. A simple calculation
shows that this condition is equivalent to the inequality $A_{+}<A_{-}$.

Let $V$ be a sufficiently small neighborhood of $p_{0}$ in $H$, and for any
distance curve $C(u,v)$ with $(u,v,0)\in V$ and for any $*\in\\{+,-\\}$, we
denote the intersection point of $C(u,v)$ with $H_{*}$ by $q_{*}(u,v)$, and
denote the arc of $C(u,v)$ between $q_{-}(u,v)$ and $q_{+}(u,v)$ by
$\Gamma(u,v)$. Applying the computations in the proof of Lemma 2.14, we have
that

$q_{*}(u,v)=\left(\sqrt{(z_{*}(u,v))^{2}+1}u,\sqrt{(z_{*}(u,v))^{2}+1}v,z_{*}(u,v)\right)$

for some $z_{*}(u,v)\in\mathbb{R}$, where the value of this expression is a
continuous function of $(u,v)$. In the following, we compute $z_{*}(u,v)$.
Substituting the coordinates of $q_{*}(u,v)$ into the equation of $H_{*}$, we
obtain the equation

$A_{*}(\sqrt{(z_{*}(u,v))^{2}+1}u-x_{0})+B_{*}\sqrt{(z_{*}(u,v))^{2}+1}v=z_{*}(u,v)$

for $z_{*}(u,v)$. After rearranging and raising to square, this equation
becomes a quadratic equation for $z_{*}(u,v)$, with solutions

$\frac{A_{*}x_{0}\pm\sqrt{(A_{*}^{2}x_{0}^{2}+1)(A_{*}u+B_{*}v)^{2}-(A_{*}u+B_{*}v)^{4}}}{(A_{*}u+B_{*}v)^{2}-1}$

in $V$, if $|A_{*}x_{0}|\neq 1$. The solutions are continuous functions of
$(u,v)$; thus, the condition that $z_{*}(x_{0},0)=0$ yields that for
$*\in\\{+,-\\}$,

(2)
$z_{*}(u,v)=\frac{A_{*}x_{0}-(A_{*}u+B_{*}v)\sqrt{(A_{*}^{2}x_{0}^{2}+1)-(A_{*}u+B_{*}v)^{2}}}{(A_{*}u+B_{*}v)^{2}-1}.$

Let $\Gamma^{\prime}(u,v)=\sigma_{L,H}(\Gamma(u,v))$. Then by Remark 2.15, the
endpoints of $\Gamma^{\prime}(u,v)$ are

$\left(\sqrt{z_{s}(u,v)^{2}+1}u,\sqrt{z_{s}(u,v)^{2}+1}v,\pm
z_{s}(u,v)\right),$

where

$z_{s}(u,v)=\frac{z_{+}(u,v)-z_{-}(u,v)}{\sqrt{(z_{+}(u,v))^{2}+1}\sqrt{(z_{-}(u,v))^{2}+1}+1+z_{+}(u,v)z_{-}(u,v)}.$

We note that the value of $z_{s}(u,v)$ is positive if and only if
$z_{+}(u,v)>z_{-}(u,v)$.

Consider the surface defined by

(3)
$r(u,v)=\left(\sqrt{z_{s}(u,v)^{2}+1}u,\sqrt{z_{s}(u,v)^{2}+1}v,z_{s}(u,v)\right)$

and satisfying $r(x_{0},0)=p_{0}$. We note that if this surface is convex,
then its Gaussian curvature at every point is nonnegative. Thus, to show that
it is not convex, it is sufficient to find values of the parameters such that
the point $p_{0}$ is a hyperbolic point of the surface, i.e., the Gaussian
curvature at this point is negative (see [6]).

Consider the quantities $e(u,v)=\langle
N(u,v),r^{\prime\prime}_{uu}(u,v)\rangle$, $f(u,v)=\langle
N(u,v),r^{\prime\prime}_{uv}(u,v)\rangle$ and $g(u,v)=\langle
N(u,v),r^{\prime\prime}_{vv}(u,v)\rangle$, where
$N(u,v)=\frac{r^{\prime}_{u}(u,v)\times
r^{\prime}_{v}(u,v)}{|r^{\prime}_{u}(u,v)\times r^{\prime}_{v}(u,v)|}$. The
Gaussian curvature $K(u,v)$ of the surface at $(u,v)$ is equal to the value of
$\frac{eg-f^{2}}{|r_{u}\times r_{v}|^{2}}$ at $(u,v)$. Thus, to show that
$K(x_{0},0)$ is negative, it suffices to show that the quantity

$F(u,v)=\langle r_{u}\times r_{v},r_{uu}\rangle\cdot\langle r_{u}\times
r_{v},r_{vv}\rangle-\left(\langle r_{u}\times r_{v},r_{uv}\rangle\right)^{2}$

is negative at $(x_{0},0)$ for certain values of $A_{*},B_{*}$ with
$*\in\\{+,-\\}$.

A computation using the Maple 18.00 software yields that if $0\leq A_{+}\leq
A_{-}$, then

(4)
$F(x_{0},0)=\frac{x_{0}^{2}}{64}(B_{+}+B_{-})(3A_{-}B_{-}-A_{-}B_{+}+A_{+}B_{-}-3A_{+}B_{+})\\\
(3A_{-}^{3}+3A_{-}^{2}A_{+}-3A_{-}A_{+}^{2}-3A_{-}B_{-}^{2}-2A_{-}B_{-}B_{+}+A_{-}B_{+}^{2}-3A_{+}^{3}-A_{+}B_{-}^{2}+2A_{+}B_{-}B_{+}+3A_{+}B_{+}^{2}).$

Thus, choosing the values $A_{-}=3$, $A_{+}=2$, $B_{-}=1$, and $B_{+}=3$, we
get $F(x_{0},0)=-139x_{0}^{2}$.

Now we construct $K$. Fix some sufficiently small angular distance
$\varphi>0$, and rotate the closed halfspaces with equations $3(x-x_{0})+y\leq
z$ and $2(x-x_{0})+3y\geq z$ about $H_{0}$ by $\varphi>0$, respectively, where
we choose the directions of the rotations differently for the two halfspaces
such that $p_{0}$ is contained in the interiors of both rotated copies. We
denote the intersection of the two rotated closed halfspaces by $D$. Note that
by our construction, $D$ is spherically convex, but $\sigma_{L,H}(D)$ is not.
Next, we rotate $D$ about $H_{0}$ such that the $z$-axis intersects the
rotated copy of $D$ in a closed segment symmetric to $o$, and denote the
rotated copy by $D^{\prime}$. Finally, we define $K$ as the intersection of
$D^{\prime}$ and its reflected copy about $o$. Then $K$ is an $o$-symmetric
spherically convex body with the property that $\sigma_{L,H}(K)$ is not
spherically convex.

To construct a spherically convex body in dimension $n>3$, we can extend the
two lunes defining $K$ to any higher-dimensional spherical space in a natural
way. ∎

###  2.4. Monotonicity of the perimeter under Steiner symmetrization

Here again we use the coordinate system introduced in the last paragraph of
Subsection 2.1. Our main result in this subsection is the following.

######  Theorem 2.18.

Let $K$ be a spherically convex set contained in $S$ satisfying the angular
monotonicity property. Then

$\operatorname{perim}(\sigma_{L,H}(K))\leq\operatorname{perim}(K).$

For the proof of Theorem 2.18, we need Lemma 2.19.

######  Lemma 2.19.

Let $p,q\in\operatorname{\mathbb{S}}^{2}$ be distinct, nonantipodal points.
Consider the ellipse

$E=\\{x\in\operatorname{\mathbb{S}}^{2}:d_{S}(p,x)+d_{S}(x,q)\leq D\\}$

with foci $p,q$, for some $0<D<2\pi$. Then $E$ is convex if and only if
$D\leq\pi$. Furthermore, if $D=\pi$, then $E$ is the closed hemisphere with
the midpoint $m$ of the segment $[p,q]_{s}$ as center.

Figure 2. An illustration for the proof of Lemma 2.19.

###### Proof.

First, assume that $D<\pi$, and let $\bar{E}$ denote the closed hemisphere
centered at $m$. Observe that for any $w\in\operatorname{bd}(\bar{E})$, we
have $d_{S}(w,p)+d_{S}(w,q)=d_{S}(w,-p)$ \+ $d_{S}(w,-q)$ by symmetry,
yielding $d_{S}(w,p)+d_{S}(w,q)=\pi$. Thus, $E$ is contained in the interior
of $\bar{E}$.

Let $x,y\in E$, and let $L$ be the greatcircle through $x,y$. Note that $x\in
E$ implies $-x\notin E$, which yields that $x,y$ are not antipodal points. Let
$z$ be any relative interior point of $[x,y]_{s}$. We show that $z\in E$.

Assume that $L$ does not separate $p$ and $q$. Let $q^{\prime}$ denote the
reflection of $q$ about the greatcircle $L$. Then the greatcircle through
$[p,z]_{s}$ intersects $[x,q^{\prime}]_{s}$ or $[y,q^{\prime}]_{s}$ at a point
$z^{\prime}$. Without loss of generality, we may assume that
$z^{\prime}\in[x,q^{\prime}]_{s}$ (see Figure 2). Then by the triangle
inequality, we have

$d_{S}(p,z)+d_{S}(z,q)=d_{S}(p,z)+d_{S}(z,q^{\prime})\leq
d_{S}(p,z)+d_{S}(z,z^{\prime})+d_{S}(z^{\prime},q^{\prime})=d_{S}(p,z^{\prime})+d_{S}(z^{\prime},q^{\prime})\leq\\\
\leq
d_{S}(p,x)+d_{S}(x,z^{\prime})+d_{S}(z^{\prime},q^{\prime})=d_{S}(p,x)+d_{S}(x,q^{\prime})=d_{S}(p,x)+d_{S}(x,q)\leq
D.$

This yields that $z\in E$. Now, if $L$ separates $p$ and $q$, then a slight
modification of this argument proves the same statement, implying that $E$ is
convex.

The remaining cases follow from the observation that the interior of the
ellipse with foci $p,q$ and parameter $D$ is the complement of the ellipse
with foci $-p,-q$ and parameter $2\pi-D$. ∎

###### Proof of Theorem 2.18.

Let $\underline{\theta},\overline{\theta}$ denote the minimum and the maximum
of the $\theta$-coordinates of the points of $K$, respectively. Since the
perimeter of a spherically convex set is a continuous function of the set, we
can assume that the boundary of $K$ is $C^{2}$-class, and there are unique
points of $K$ whose $\theta$-coordinates are minimal or maximal. Then
$\operatorname{bd}(K)$ can be written as the union of two $C^{2}$-class curves
$\Gamma_{i}:[\underline{\theta},\overline{\theta}]\to S$ ($i=1,2$),
parametrized by the value of the $\theta$-coordinates of the points. We choose
the indices of the curves such that for
$\underline{\theta}<\theta<\overline{\theta}$, the $\varphi$-coordinate of
$\Gamma_{2}(\theta)$ is larger than that of $\Gamma_{1}(\theta)$. Thus,

$\operatorname{perim}(K)=\int_{\underline{\theta}}^{\overline{\theta}}||\dot{\Gamma}_{1}(\theta)||\,d\theta+\int_{\underline{\theta}}^{\overline{\theta}}||\dot{\Gamma}_{2}(\theta)||\,d\theta,$

where $||\cdot||$ denotes the Euclidean norm. Similarly,
$\operatorname{bd}(\sigma_{L,H}(K))$ can be written as the union of two
$C^{2}$-class curves $\Delta_{i}:[\underline{\theta},\overline{\theta}]\to S$
($i=1,2$), parametrized by the value of the $\theta$-coordinates of the points
such that for $\underline{\theta}<\theta<\overline{\theta}$, the
$\varphi$-coordinate of $\Delta_{2}(\theta)$ is larger than that of
$\Delta_{1}(\theta)$. Hence,

$\operatorname{perim}(K)=\int_{\underline{\theta}}^{\overline{\theta}}||\dot{\Delta}_{1}(\theta)||\,d\theta+\int_{\underline{\theta}}^{\overline{\theta}}||\dot{\Delta}_{2}(\theta)||\,d\theta.$

We intend to show that for any
$\underline{\theta}\leq\theta\leq\overline{\theta}$, we have

$||\dot{\Gamma}_{1}(\theta)||+||\dot{\Gamma}_{2}(\theta)||\geq||\dot{\Delta}_{1}(\theta)||+||\dot{\Delta}_{2}(\theta)||.$

To do this, it suffices to show that if $0\leq\theta\leq\theta^{\prime}$ or
$\theta\leq\theta^{\prime}\leq 0$ are sufficiently close, then

(5)
$d_{S}(\Gamma_{1}(\theta^{\prime}),\Gamma_{1}(\theta))+d_{S}(\Gamma_{2}(\theta^{\prime}),\Gamma_{2}(\theta))\geq
d_{S}(\Delta_{1}(\theta^{\prime}),\Delta_{1}(\theta))+d_{S}(\Delta_{2}(\theta^{\prime}),\Delta_{2}(\theta)).$

Without loss of generality, we may assume that $0\leq\theta<\theta^{\prime}$.
For $i=1,2$, let $p_{i}=\Gamma_{i}(\theta)$,
$p_{i}^{\prime}=\Gamma_{i}(\theta^{\prime})$, $q_{i}=\Delta_{i}(\theta)$ and
$q_{i}^{\prime}=\Delta_{i}(\theta^{\prime})$. Note that by the definition of
$\sigma_{L,H}$, we have
$\varphi_{p_{2}}-\varphi_{p_{1}}=\varphi_{q_{2}}-\varphi_{q_{1}}$ and
$\varphi_{p_{2}^{\prime}}-\varphi_{p_{1}^{\prime}}=\varphi_{q_{2}^{\prime}}-\varphi_{q_{1}^{\prime}}$.
Furthermore, by the angular monotonicity property, the first quantity is not
smaller than the second quantity. Similar to the proof of Theorem 2.12, we may
assume that $p_{1}=q_{1}$, $p_{2}=q_{2}$ and $p_{1}^{\prime}=p_{2}^{\prime}$,
implying also that $q_{1}^{\prime}=q_{2}^{\prime}$, with
$\varphi_{q_{1}^{\prime}}=\varphi_{q_{2}^{\prime}}=0$. Since
$0\leq\theta<\theta^{\prime}$ is sufficiently small, we can assume that
$d_{s}(q_{1},q_{1}^{\prime})+d_{s}(q_{2},q_{2}^{\prime})=D<\pi$. Thus, by
Lemma 2.19 the ellipse $E$ with foci $p_{1},p_{2}$ and parameter $D$ is
convex, and by symmetry, the distance curve $C_{\theta^{\prime}}$ supports $E$
at $q_{1}^{\prime}$. Hence, $p_{1}^{\prime}=p_{2}^{\prime}$ does not lie in
the interior of $E$, implying (5). ∎

###  2.5. Monotonicity of the diameter under Steiner symmetrization

######  Theorem 2.20.

Let $K$ be a spherically convex set contained in $S$ satisfying the angular
monotonicity property. Then

$\operatorname{diam}(\sigma_{L,H}(K))\leq\operatorname{diam}(K).$

###### Proof.

Let $[p,q]_{s}$ be a diameter of $\sigma_{L,H}(K)$, and let $C_{p}$ and
$C_{q}$ denote the distance curves with axis $L$ passing through $p$ and $q$,
respectively. Without loss of generality, we may assume that $\varphi_{p}\geq
0$ and $\varphi_{q}\leq 0$, and $|\varphi_{p}|\geq|\varphi_{q}|$. Let
$p^{\prime}$ and $q^{\prime}$ denote the reflections of $p,q$ about $H$,
respectively. Then
$p,q,p^{\prime},q^{\prime}\in\operatorname{bd}(\sigma_{L,H}K)$, and
$[p^{\prime},q^{\prime}]_{s}$ is also a diameter of $\sigma_{L,H}(K)$.

Let $\bar{p},\bar{p}^{\prime}\in\operatorname{bd}(K)$ denote the ‘preimages’
of $p,p^{\prime}$, respectively, under $\sigma_{L,H}$; that is, the rotation
about the poles that maps $C_{p}\cap\sigma_{L,H}(K)$ into $C_{p}\cap K$ maps
$p$ into $\bar{p}$ and $p^{\prime}$ into $\bar{p}^{\prime}$. We define the
points $\bar{q},\bar{q}^{\prime}\in\operatorname{bd}(K)$ similarly. To prove
Theorem 2.20, it suffices to show that

(6)
$d_{s}(p,q)=d_{s}(p^{\prime},q^{\prime})\leq\max\\{d_{s}(\bar{p},\bar{q}),d_{s}(\bar{p}^{\prime},\bar{q}^{\prime})\\}.$

Let $y,y^{\prime}$ denote the points of $C_{q}$ satisfying
$\varphi_{y}=\varphi_{\bar{p}}$ and
$\varphi_{y^{\prime}}=\varphi_{\bar{p}^{\prime}}$, and let $z,z^{\prime}$
denote the points of the spherical circle $\hat{C}_{q}$ containing $C_{q}$ and
opposite to $y,y^{\prime}$, respectively. Note that for any point
$u\in\hat{C}_{q}$, we have $d_{s}(\bar{p},y)\leq d_{s}(\bar{p},u)\leq
d_{s}(p,z)$ with equality on the left if $u=y$ and on the right if $u=z$, and
as we move $u$ on $\hat{C}_{q}$ towards $z$, then $d_{s}(p,u)$ strictly
increases. The same statement holds if we replace $\bar{p},y,z$ by
$\bar{p}^{\prime},y^{\prime},z^{\prime}$, respectively. Since
$z,z^{\prime}\notin S$, this yields that if $G\subset S$ is an arc of angular
length $2|\varphi_{q}|$ with endpoints $u,v$, where
$\varphi_{u}\geq\varphi_{v}$, then $\max\\{d_{s}(p,v),d_{s}(q,u)\\}$ is
minimal if the midpoints of this arc and the arc of $C_{p}$ with endpoints
$\bar{p},\bar{p^{\prime}}$ have the same $\varphi$-coordinate. This proves
(6). ∎

###  2.6. Approximating spherical caps by subsequent Steiner symmetrizations

Gross [10] proved that for any convex body $K$ in $\mathbb{R}^{n}$, a
Euclidean ball can be approximated arbitrarily well, measured in Hausdorff
distance, by applying subsequent Steiner symmetrizations to $K$. Our next
result is an analogue of this statement for centrally symmetric convex sets
and spherical Steiner symmetrization.

######  Theorem 2.21.

Let $K\in\mathcal{K}_{S}$ be a $c$-symmetric convex disk, and let
$\mathcal{F}_{K}$ denote the family of $c$-symmetric convex disks that can be
obtained from $K$ by finitely many subsequent Steiner symmetrizations
$\sigma_{L,H}$ with $L\cap H=\\{c\\}$. Then there is a sequence
$\\{K_{m}\\}_{m=1}^{\infty}$ of $c$-symmetric convex disks with
$K_{m}\in\mathcal{F}_{K}$ such that $K_{m}\to D_{c}(K)$ in the Hausdorff
metric as $m\to\infty$, where $D_{c}(K)$ is the spherical cap centered at $c$
with $\operatorname{area}(K)=\operatorname{area}(D_{c}(K))$.

We note that an important part of the proof of Theorem 2.21 is that the image
of any $c$-symmetric convex disk under any Steiner symmetrization
$\sigma_{L,H}$ with $H\cap L=\\{c\\}$ is a $c$-symmetric convex disk.

###### Proof.

Observe that a $c$-symmetric convex disk satisfies the angular monotonicity
property for any Steiner symmetrization $\sigma_{L,H}$ with $L\cap H=\\{c\\}$,
implying that for such a Steiner symmetrization $\sigma_{L,H}(K)$ is also a
$c$-symmetric convex disk. Furthermore, Blaschke’s selection theorem holds for
the space of spherically convex bodies in $\operatorname{\mathbb{S}}^{2}$
(see, for example, [12, page 88]). From here the proof of [12, Theorem 9.1]
can be modified in a straightforward way to obtain the result. ∎

######  Definition 2.22.

A sequence of Steiner symmetrizations $\\{\sigma_{L_{i},H_{i}}\\}_{i=1}^{k}$
in $\operatorname{\mathbb{S}}^{2}$ is called _applicable_ to a convex disk
$K\subset\operatorname{\mathbb{S}}^{2}$ if:

  * (i)

$K$ is contained in the open hemisphere centered at the intersection point of
$H_{1}$ and $L_{1}$, and it satisfies the angular monotonicity property with
respect to $\sigma_{L_{1},H_{1}}$;

  * (ii)

for any $i=1,2,\ldots,k-1$,
$\sigma_{L_{i-1},H_{i-1}}(\ldots\sigma_{L_{1},H_{1}(K)})$ is contained in the
open hemisphere centered at the intersection point of $L_{i}\cap H_{i}$, and
it satisfies the angular monotonicity property with respect to
$\sigma_{L_{i},H_{i}}$.

######  Theorem 2.23.

Let $K\subset\operatorname{\mathbb{S}}^{2}$ be a convex disk of diameter less
than $\frac{\pi}{2}$. Let $\mathcal{F}_{K}$ denote the family of the congruent
copies of images of $K$ under finite sequences of Steiner symmetrizations
applicable to $K$. There exists a sequence
$\\{K_{m}\\}_{m=1}^{\infty}\subset\mathcal{F}(K)$ such that $K_{m}\to D(K)$ in
the Hausdorff metric, where $D(K)$ is a spherical cap with
$\operatorname{area}(D(K))=\operatorname{area}(K)$.

The proof is based on the following lemma.

######  Lemma 2.24.

Let $K$ be a convex disk of diameter less than $\frac{\pi}{2}$ with circumdisk
$D$ and circumradius $\rho<\frac{\pi}{2}$. Assume that some
$p\in\operatorname{bd}(D)$ is at distance at least $\varepsilon>0$ from $K$.
Then there is some lower semicontinuous function
$\delta=\delta(\rho,\varepsilon)>0$, such that there is a finite sequence of
Steiner symmetrizations $\\{\sigma_{L_{i},H_{i}}\\}_{i=1}^{k}$ applicable to
$K$ satisfying the property that
$\sigma_{L_{k},H_{k}}(\ldots\sigma_{L_{1},H_{1}(K)})$ is contained in the
spherical cap of radius $\rho-\delta$ and concentric with $D$.

###### Proof of Lemma 2.24.

By the conditions of the lemma, there is a closed circle arc $G_{0}$ in
$\operatorname{bd}(C)$ such that its distance from $K$ is at least
$\varepsilon>0$. Let $G_{i}\subset\operatorname{bd}(D)$, $i=1,2,\ldots,k$, be
a family of congruent copies of $G$ which satisfies
$\bigcup_{i=0}^{k}G_{i}=\operatorname{bd}(D)$. Let $H_{i}$ be the line through
the center $c$ of $D$ such that the reflected copy of $G$ to $H_{i}$ is
$G_{i}$. By compactness, there is a value $\delta_{1}>0$ such that for any
line $L_{1}$ perpendicular to $H_{1}$ and $L_{1}\cap H_{1}\subset D$, if
$\sigma_{L_{1},H_{1}}$ is applicable to $K$, then the distance of both
$G=G_{0}$ and $G_{1}$ from $\sigma_{L_{1},H_{1}}(K)$ is at least $\delta_{1}$.
Similarly, if $K^{\prime}\subset D$ has the property that its distance from
both $G_{0}$ and $G_{1}$ is at least $\delta_{1}$, then there is some
$\delta_{2}>0$ such that for any line $L_{2}$ perpendicular to $H_{2}$ and
$L_{2}\cap H_{2}\subset D$, if $\sigma_{L_{2},H_{2}}$ is applicable to
$K^{\prime}$, then the distance of $G_{0},G_{1},G_{2}$ from
$\sigma_{L_{2},H_{2}}(K^{\prime})$ is at least $\delta_{2}$. In general, if
for any $1\leq i\leq k-1$, some convex disk $K^{\prime}\subset D$ has the
property that its distance from $G_{0},G_{1},\ldots,G_{i}$ is at least
$\delta_{i}>0$ from $K^{\prime}$, then there is some $\delta_{i+1}>0$ such
that for any line $L_{i+1}$ perpendicular to $H_{i+1}$ and $L_{i+1}\cap
H_{i+1}\subset D$, if $\sigma_{L_{i+1},H_{i+1}}$ is applicable to
$K^{\prime}$, then the distance of $G_{0},G_{1},\ldots,G_{i+1}$ from
$\sigma_{L_{i+1},H_{i+1}}(K^{\prime})$ is at least $\delta_{i+1}$.

Thus, it remains to show that there is a sequence $\sigma_{L_{i}\cap H_{i}}$
of Steiner symetrizations applicable to $K$ such that $L_{i}\cap H_{i}\subset
D$. By continuity, without loss of generality we may assume that $K$ is
smooth. Let $[p,q]_{s}=K\cap H_{1}$, and let $\Gamma_{1}$ and $\Gamma_{2}$
denote the two curves of $\operatorname{bd}(K)$ that connect $p,q$. Note that
since $D$ is the circumdisk of $K$, the circumcenter $c$ of $K$ lies on
$[p,q]_{s}$. For any $z\in[p,q]_{s}$, let $L_{z}$ denote the line through $z$
that is perpendicular to $H_{1}$. Let $z\in[p,q]_{s}\setminus\\{p,q\\}$. Then
$L_{z}$ intersects $\Gamma_{1}$ and $\Gamma_{2}$ in unique points, which we
denote by $w_{1}(z)$ and $w_{2}(z)$, respectively. Let $K(z)$ denote the
closure of the component of $K\setminus L_{z}$ containing $q$, and for
$i=1,2$, let $\alpha_{i}$ denote the angle of $K(z)$ at $w_{i}(z)$. Observe
that if $z$ is close to $p$, then $\alpha_{1}(z)+\alpha_{2}(z)$ is close to
$2\pi$, and if $z$ is close to $q$, then $\alpha_{1}(z)+\alpha_{2}(z)$ is
close to $0$. Thus, by continuity, there is some
$z\in[p,q]_{s}\setminus\\{p,q\\}$ such that $\alpha_{1}(z)+\alpha_{2}(z)=\pi$.
But then $K$ is contained in the open hemisphere centered at $z$, and by Lemma
2.9, $K$ satisfies the angular monotonicity with respect to
$\sigma_{L_{z},H_{1}}$. As $z\in\sigma_{L_{z},H_{1}}(K)$ due to the
construction, by continuing this process we can construct the desired sequence
of Steiner symmetrizations.

Finally, the lower semicontinuity of the value $\delta(\varepsilon,\rho)$
follows from the method of construction. ∎

Before the proof of Theorem 2.23, we recall that every spherically convex body
$K$ has a unique circumball which depends continuously on $K$ with respect to
Hausdorff distance.

###### Proof of Theorem 2.23.

For any convex disk $K$, let $\rho(K)$ denote the circumradius of $K$, and let
$\varepsilon(K)$ denote the Hausdorff distance between $K$ and its circumdisk.
Let $\rho=\inf\\{\rho(K^{\prime}):K^{\prime}\in\mathcal{F}_{K}\\}$. By
Blaschke’s selection theorem, there is a convex disk $K^{\prime}$ with
$\rho(K^{\prime})=\varepsilon$ such that $K^{\prime}$ is the limit of a
sequence $\\{K_{m}\\}$ of elements of $\mathcal{F}_{K}$. If $K^{\prime}$ is a
spherical cap, then we are done. Assuming that $K^{\prime}$ is not a spherical
cap yields that $\varepsilon(K^{\prime})>0$. Let
$\varepsilon=\varepsilon(K^{\prime})$ and $0<\delta<\delta(\rho,\varepsilon)$,
where $\delta(\rho,\varepsilon)$ is the function whose existence is proved in
Lemma 2.24. Observe that $K_{m}\to K^{\prime}$ implies that
$\rho(K_{m})\to\rho(K^{\prime})$ and
$\varepsilon(K_{m})\to\varepsilon(K^{\prime})$. Thus, by the lower
semicontinuity of $\delta(\rho,\varepsilon)$, for all sufficiently large $m$
we have $\delta(\rho(K_{m}),\varepsilon(K_{m}))>\delta$ and
$0\leq\rho(K_{m})-\rho<\delta$. But then Lemma 2.24 yields the existence of a
finite sequence $\sigma_{L_{i},H_{i}}$ ($i=1,2,\ldots,k$) of Steiner
symmetrizations applicable to $K_{m}$ such that
$\sigma_{L_{k},H_{k}}(\ldots\sigma_{L_{1},H_{1}(K_{m})})$ is contained in a
spherical cap of radius

$\rho(K_{m})-\delta(\rho(K_{m}),\varepsilon(K_{m}))<(\rho+\delta)-\delta=\rho,$

contradicting the definition of $\rho$. ∎

##  3\. Applications of spherical Steiner symmetrization

Let us recall that $S$ is an open hemisphere of
$\operatorname{\mathbb{S}}^{2}$ centered at $c$. In the following, we denote
the family of $c$-symmetric convex disks in $S$ by $\mathcal{K}^{\rm
sym}_{S}$, and the family of Steiner symmetrizations $\sigma_{L,H}$ with
$L\cap H=\\{c\\}$ by $\Sigma_{c}$. We observe that by Remark 2.11 and Theorem
2.12, for any $K\in\mathcal{K}^{\rm sym}_{S}$ and $\sigma_{L,H}\in\Sigma_{c}$,
we have $\sigma_{L,H}(K)\in\mathcal{K}^{\rm sym}_{S}$.

We start this section with a general theorem about finding optimal values of
functions defined on $\mathcal{K}^{\rm sym}_{S}$.

######  Theorem 3.1.

Assume that $F:\mathcal{K}^{\rm sym}_{S}\to(0,\infty)$ is nondecreasing (resp.
nonincreasing) under symmetrizations $\sigma_{L,H}\in\Sigma_{c}$, and it is
upper semicontinuous (resp. lower semicontinuous) with respect to the
Hausdorff metric. Then for every $K\in\mathcal{K}^{\rm sym}_{S}$, we have

$F(K)\leq F(K_{c}^{*})\quad(\hbox{resp. }F(K)\geq F(K_{c}^{*})),$

where $K_{c}^{*}$ is the spherical cap centered at $c$ with
$\operatorname{area}(K_{c}^{*})=\operatorname{area}(K)$.

###### Proof.

As in the proof of Theorem 2.21, let $\mathcal{F}_{K}$ denote the family of
$c$-symmetric convex disks that can be obtained from $K$ by finitely many
subsequent Steiner symmetrizations in $\Sigma_{c}$. By Theorem 2.21, there
exists a sequence $\\{K_{m}\\}\subset\mathcal{F}_{K}$ such that $K_{m}\to
K_{c}^{*}$ with respect to the Hausdorff metric. Thus, if $F$ is nondecreasing
under symmetrizations $\sigma_{L,H}\in\Sigma_{c}$, then $F(K)\leq F(K_{m})$
for any value of $m$. On the other hand, from the upper semicontinuity of $F$
it follows that

$\limsup_{m\to\infty}F(K_{m})\leq F(K_{c}^{*}),$

implying Theorem 3.1 in this case. If $F$ is nonincreasing under
symmetrizations $\sigma_{L,H}\in\Sigma_{c}$ and it is lower semicontinuous, we
may apply a similar argument. ∎

Theorem 3.1, combined with Theorems 2.18 and 2.20, readily yields the
isoperimetric and the isodiametric inequalities for $c$-symmetric convex
disks.

######  Corollary 3.2.

Among elements of $\mathcal{K}^{\rm sym}_{S}$ of a given area, spherical caps
centered at $c$ have minimal perimeter and diameter.

###  3.1. A variant of a theorem of Sas for $c$-symmetric convex disks in
$\operatorname{\mathbb{S}}^{2}$

The main concept of this subsection is the following.

######  Definition 3.3.

Let $K\subset\mathbb{S}^{n}$ be a spherically convex body and let $N\geq n+1$.
Then by $A_{N}(K)$ we denote the supremum of the volumes of all spherically
convex polytopes in $K$ with at most $N$ vertices.

We note that by compactness, the value $A_{N}(K)$ is attained at some
spherically convex polytope in $K$ with $N$ vertices.

The minimum of this quantity has been studied extensively in the Euclidean
setting. More specifically, Blaschke [4, pp. 49–53] proved that for any convex
body $K$ in $\mathbb{R}^{2}$, there exists a triangle $T\subset K$ such that

(7) $\operatorname{area}(T)\geq\frac{3\sqrt{3}}{4\pi}\operatorname{area}(K),$

with equality if and only if $K$ is an ellipse. Sas [20] later extended this
result to all convex polygons, showing that there exists an $N$-gon
$P_{N}\subset K$ such that

(8)
$\operatorname{area}(P_{N})\geq\frac{N}{2\pi}\sin\frac{2\pi}{N}\operatorname{area}(K)$

with equality if and only if $K$ is an ellipse.

Apart from the equality case, this result was extended by Macbeath [15] to
polytopes with $N$ vertices contained in a convex body $K$ in
$\mathbb{R}^{n}$.

In our next result, we prove an analogue of the result of Sas [20] for
$c$-symmetric convex disks on $\operatorname{\mathbb{S}}^{2}$.

######  Theorem 3.4.

Let $K\in\mathcal{K}^{\rm sym}_{S}$ and $N\geq 3$, and let $B_{c}(r)$ denote
the spherical cap centered at $c$ that satisfies
$\operatorname{area}(K)=\operatorname{area}(B_{c}(r))$. Then we have

(9) $A_{N}(K)\geq A_{N}(B_{c}(r))=C(r,N),$

where

(10)
$C(r,N)=\frac{2N\arctan\left(\dfrac{1+\cos\frac{2\pi}{N}}{\sin\frac{2\pi}{N}}\cos
r\right)-(N-2)\pi}{2\pi(1-\cos r)}.$

We start the proof with a lemma.

######  Lemma 3.5.

For any $K\in\mathcal{K}^{\rm sym}_{S}$ and any convex $N$-gon $P$ of maximal
area in $K$, where $N\geq 3$, we have $c\in P$.

###### Proof.

Consider an $N$-gon $P$ contained in $K$ that does not contain $c$. Then there
is a greatcircle $L$ through $c$ disjoint from $P$. Let the endpoints of
$L\cap K$ be denoted by $p$ and $q$. Since $K\in\mathcal{K}^{\rm sym}_{S}$,
there is a greatcircle $L$ through $c$, and two distance curves $C_{p}$ and
$C_{q}$ with axis $L$ and that pass through $p$ and $q$, respectively, which
touch $\operatorname{bd}(K)$. Here, without loss of generality, we may assume
that $L$ is the line $L$ in the coordinate system in the last paragraph of
Subsection 2.1. We denote by $K_{0}$ the intersection of $K$ with the closed
hemisphere of $\operatorname{\mathbb{S}}^{2}$ bounded by $L$ and containing
$P$.

Let $C$ be any distance curve with axis $L$ that intersects $K$. Then $C$
intersects $[p,q]_{s}$, and by compactness, there is some $\varepsilon>0$ such
that for any such $C$, the angular length of $C\cap K_{0}$ is greater by
$\varepsilon$ than the angular length of $C\cap P$. Let $P^{\prime}$ be the
rotated copy of $P$ about the poles of $\operatorname{\mathbb{S}}^{2}$ by
$\varepsilon$ towards $L$. Then $P^{\prime}\subseteq K_{0}\subset K$, and at
least one vertex of $P^{\prime}$ is contained in the interior of $K$. This
yields that the area of $P$ is not maximal. ∎

###### Proof of Theorem 3.4.

First, note that by the spherical variant of Dowker’s theorem (c.f., e.g.,
[7]), any $N$-gon of maximal area inscribed in $B_{c}(r)$ is a regular
$N$-gon. Using spherical trigonometry and Girard’s theorem, one can compute
the area $C(r,N)$ of such a polygon, obtaining the formula in (10). This
proves the equality in (9). Thus, we need to prove the inequality in (9). To
do so, we choose an arbitrary symmetrization $\sigma_{L,H}\in\Sigma_{c}$, and
observe that by Theorem 3.1, it suffices to show that for any
$K\in\mathcal{K}^{\rm sym}_{S}$ and $m\geq 3$, we have $A_{N}(K)\geq
A_{N}(\sigma_{L,H}(K))$.

Figure 3. An illustration for the proof of Theorem 3.4 with $N=6$. The dotted
arcs denote the chords of $K$ on the distance curves $C_{i}$\- The dashed
polygonal curves denote the boundaries of the spherical inscribed polygons $P$
and $Q$.

Set $K^{\prime}=\sigma_{L,H}(K)$, and let $P^{\prime}$ be an $N$-gon contained
in $K^{\prime}$ of area $A_{N}(K^{\prime})$. Then $P^{\prime}$ is inscribed in
$K^{\prime}$, and since $K^{\prime}$ is axially symmetric to $H$, the
reflected copy $Q^{\prime}$ of $P$ about $H$ is also a maximal area $N$-gon in
$K^{\prime}$. Let $p_{1}^{\prime},p_{2}^{\prime},\ldots,p_{N}^{\prime}$ be the
vertices of $P^{\prime}$ in counterclockwise order, where we set
$p_{N+1}^{\prime}=p_{1}^{\prime}$. We follow the idea of Macbeath in [15], and
for every $p_{i}^{\prime}$ we define $C_{i}$ as the distance curves with axis
$L$, passing through $p_{i}^{\prime}$. Then $C_{i}\cap K^{\prime}$ is a circle
arc with endpoints $p_{i}^{\prime}$ and $q_{i}^{\prime}$, where
$q_{i}^{\prime}$ is the reflection of $p_{i}^{\prime}$ to $H$, and thus, it is
a vertex of $Q^{\prime}$. The curve $C_{i}$ intersects $K$ in a rotated copy
of the circle arc $C_{i}\cap K^{\prime}$. We denote the endpoints of this arc
by $p_{i}$ and $q_{i}$, where the notation is chosen such that the rotation
that moves $p_{i}^{\prime}$ into $p_{i}$ moves $q_{i}^{\prime}$ into $q_{i}$
(see Figure 3).

In the remaining part, we show that

$\operatorname{area}(P)+\operatorname{area}(Q)\geq\operatorname{area}(P^{\prime})+\operatorname{area}(Q^{\prime}).$

For $i=1,2,\ldots,N$, we define the quantities $A_{i}^{\prime}$ as follows.
Consider the case that $[p_{i}^{\prime},q_{i+1}^{\prime}]_{s}$ and
$[q_{i}^{\prime},p_{i+1}^{\prime}]_{s}$ do not cross. Then $A_{i}^{\prime}$
denotes the area of the region in $S$ bounded by
$[p_{i}^{\prime},q_{i+1}^{\prime}]_{s}$,
$[q_{i}^{\prime},p_{i+1}^{\prime}]_{s}$, and the circle arcs $C_{i}\cap
K^{\prime}$ and $C_{i+1}\cap K^{\prime}$. Assume that
$[p_{i}^{\prime},q_{i+1}^{\prime}]_{s}$ and
$[q_{i}^{\prime},p_{i+1}^{\prime}]_{s}$ cross at a point $x$. For $j=i,i+1$,
let $R_{j}$ denote the region bounded by $[p_{j}^{\prime},x]_{s}$,
$[q_{j}^{\prime},x]_{s}$ and $C_{j}\cap K^{\prime}$. Then one of $R_{i}$ and
$R_{i+1}$ is covered by both $P^{\prime}$ and $Q^{\prime}$, and the other one
overlaps neither $P^{\prime}$ nor $Q^{\prime}$. Let the first region be
denoted by $R_{+}$, and the second one by $R_{-}$. Then we set
$A_{i}^{\prime}=\operatorname{area}(R_{+})-\operatorname{area}(R_{-})$.
Finally, we observe that $[p_{i}^{\prime},q_{i+1}^{\prime}]_{s}$ and
$[q_{i}^{\prime},p_{i+1}^{\prime}]_{s}$ cross if and only if
$[p_{i},q_{i+1}]_{s}$ and $[q_{i},p_{i+1}]_{s}$ cross, and define the
quantities $A_{i}$ analogously to the $A_{i}^{\prime}$.

Now, we have

$\operatorname{area}(P^{\prime})+\operatorname{area}(Q^{\prime})=\sum_{i=1}^{m}A_{i}^{\prime},\quad\operatorname{area}(P)+\operatorname{area}(Q)=\sum_{i=1}^{N}A_{i}.$

Thus, it is sufficient to prove that for any value of $i$, $A_{i}^{\prime}\geq
A_{i}$, which we are going to do in the remaining part of the proof.

Consider some $i$ such that $[p_{i}^{\prime},q_{i+1}^{\prime}]_{s}$ and
$[q_{i}^{\prime},p_{i+1}^{\prime}]_{s}$ cross neither each other, nor $L$.
Then applying the idea of the proof of Theorem 2.12, we obtain that $A_{i}\geq
A_{i}^{\prime}$.

Assume that $[p_{i}^{\prime},q_{i+1}^{\prime}]_{s}$ and
$[q_{i}^{\prime},p_{i+1}^{\prime}]_{s}$ cross $L$ but not each other. Without
loss of generality, we may assume that the distance of $C_{i}$ from $L$ is not
greater than that of $C_{i+1}$. Thus, by the symmetry and angular monotonicity
property of $K$, the angular length of $C_{i}\cap K^{\prime}$ is not less than
that of $C_{i+1}\cap K^{\prime}$. We follow the idea of the proof of Theorem
2.12, and, without loss of generality, assume that $C_{i+1}\cap K^{\prime}$ is
a single point, i.e., $p_{i+1}^{\prime}=q_{i+1}^{\prime}$, which yields also
that $p_{i+1}=q_{i+1}$. But then the Lexell circle of
$[p_{i}^{\prime},q_{i}^{\prime}]_{s}$ through this point separates $C_{i+1}$
from $[p_{i}^{\prime},q_{i}^{\prime}]_{s}$, implying that
$\operatorname{area}(\operatorname{conv}_{s}\\{p_{i},q_{i},p_{i+1}\\})\geq\operatorname{area}(\operatorname{conv}_{s}\\{p_{i}^{\prime},q_{i}^{\prime},p_{i+1}^{\prime}\\})$
and thus, $A_{i}\geq A_{i}^{\prime}$.

Consider the case that $[p_{i}^{\prime},q_{i+1}^{\prime}]_{s}$ and
$[q_{i}^{\prime},p_{i+1}^{\prime}]_{s}$ cross each other, but not $L$. Without
loss of generality, we assume that $C_{i}\cap K^{\prime}$ lies in the boundary
of the region $R_{+}$ of positive weight. By Lemma 3.5 and the symmetry of
$K^{\prime}$, the distance of $C_{i}$ from $L$ is not greater than that of
$C_{i+1}$, as otherwise $c\notin P$.

Assume that $C_{i}\cap\sigma(K)$ belongs to the part with positive weight.
Then by Lemma 3.5 and the symmetry of $\sigma(K)$, $C_{i}$ is not closer to
$G$ than $C_{i}$, as otherwise $c\notin P$. Thus, as in the previous cases,
applying a suitable rotation to $[p_{i}^{\prime},p_{i+1}^{\prime}]_{s}$ and
$[q_{i}^{\prime},q_{i+1}^{\prime}]_{s}$, we may assume that
$q_{i+1}^{\prime}=p_{i+1}^{\prime}$, implying that $q_{i+1}=p_{i+1}$. As
before, we have
$\operatorname{area}(\operatorname{conv}_{s}\\{p_{i},q_{i},p_{i+1}\\})\geq\operatorname{area}(\operatorname{conv}_{s}\\{p_{i}^{\prime},q_{i}^{\prime},p_{i+1}^{\prime}\\})$,
which yields that $A_{i}\geq A_{i}^{\prime}$.

Finally, if $[p_{i}^{\prime},q_{i+1}^{\prime}]_{s}$ and
$[q_{i}^{\prime},p_{i+1}^{\prime}]_{s}$ cross each other and also $L$, to
obtain the inequality $A_{i}\geq A_{i}^{\prime}$ we can combine the arguments
in the two previous cases. ∎

######  Problem 3.6.

Remove the symmetry assumption on the convex disk $K$ in Theorem 3.4.

###  3.2. An isoperimetric-type inequality for floating areas of spherically
convex disks

Let $\mathcal{K}^{n}$ denote the set of all convex bodies in $\mathbb{R}^{n}$.
For $K\in\mathcal{K}^{n}$, the generalized Gauss curvature at
$x\in\operatorname{bd}(K)$ is denoted by $H_{n-1}(K,x)$. Note that
$H_{n-1}(K,x)$ exists for $\mathcal{H}^{n-1}$-almost all
$x\in\operatorname{bd}(K)$ [13, Lemma 2.3]. The quantity

$\operatorname{as}(K):=\int_{\operatorname{bd}(K)}H_{n-1}(K,x)^{\frac{1}{n+1}}\,d\mathcal{H}^{n-1}(x)$

is called the _affine surface area_ of $K$. It is invariant under volume-
preserving affine transformations, and it is an upper semicontinuous valuation
on the set of convex bodies in $\mathbb{R}^{n}$ which vanishes on polytopes.
For more background on affine surface area, we refer the reader to, e.g., [21,
Section 10.5].

Interestingly, the quantity $\operatorname{as}(K)$ is closely related to how
well $K$ can be approximated by polytopes, as follows. For any
$K\in\mathcal{K}^{n}$, let $\mathcal{P}_{N}(K)$ denote the set of all
polytopes contained in $K$ with at most $N$ vertices, and set

(11)
$\operatorname{dist}(K,\mathcal{P}_{N}(K)):=\inf\left\\{\operatorname{vol}_{n}(K)-\operatorname{vol}_{n}(P):\,P\in\mathcal{P}_{N}(K)\right\\}.$

By a compactness argument, there exists a _best-approximating polytope_
$\widehat{P}\in\mathcal{P}_{N}(K)$ which attains the infimum in (11).

If $K\in\mathcal{K}^{n}$ has $C^{2}$ boundary and positive Gauss curvature,
then

(12)
$\lim_{N\to\infty}\operatorname{dist}(K,\mathcal{P}_{N}(K))N^{\frac{2}{n-1}}=\frac{1}{2}{\rm
del}_{n-1}\left[{\rm as}(K)\right]^{\frac{n+1}{n-1}}$

Here the quantity $\operatorname{del}_{n-1}$ is the _Delone triangulation
number_ in $\mathbb{R}^{n}$, and it depends only on the dimension $n$. Only
the first few values of $\operatorname{del}_{n-1}$ are known; in particular,
the exact value ${\rm del}_{1}=1/6$ was stated by L. Fejes Tóth [8] and proved
by McClure and Vitale [18], and Gruber [11] proved that ${\rm
del}_{2}=1/(2\sqrt{3})$. The best known asymptotic estimate for
$\operatorname{del}_{n-1}$ is due to Mankiewicz and Schütt [16, 17], who
proved that

${\rm del}_{n-1}=\frac{n}{2\pi e}\left(1+O\left(\frac{\ln
n}{n}\right)\right).$

The spherical analogue of affine surface area for a spherically convex body
$K$ is the so-called _floating area_ of $K$, defined by Besau and Werner [3]
as

(13) $\Omega_{s}(K)=\int_{\partial
K}H_{n-1}^{\mathbb{S}^{n}}(K,x)^{\frac{1}{n+1}}\,d\mathcal{H}^{n-1}(x).$

The function $\Omega_{s}(\cdot)$ is an upper semicontinuous valuation on the
set $\mathcal{K}(\mathbb{S}^{n})$ of spherically convex bodies, and it
vanishes on spherical polytopes [3].

The following conjecture is due to Besau and Werner [3, Conjecture 7.5].

######  Conjecture 3.7 (Isoperimetric Inequality for Floating Area).

Let $K\in\mathcal{K}(\mathbb{S}^{n})$. Then
$\Omega_{s}(K)\leq\Omega_{s}(K_{s}^{*})$, where $K_{s}^{*}$ is a spherical
ball such that $\operatorname{vol}_{s}(K)=\operatorname{vol}_{s}(K_{s}^{*})$.
Equality holds if and only if $K$ is a spherical cap.

In this subsection, we prove that the inequality in Conjecture 3.7 holds for
smooth and $c$-symmetric convex disks in $\mathbb{S}^{2}$. In what follows,
given $p\in\mathbb{S}^{n}$, let
$\mathcal{K}_{p}(\mathbb{S}^{n}):=\\{K\in\mathcal{K}(\mathbb{S}^{n}):\,p\in
K,K\subset S\\}$.

######  Theorem 3.8.

Let $K\in\mathcal{K}_{c}(\mathbb{S}^{2})$ be $c$-symmetric and have
$C^{2}_{+}$ boundary. Then $\Omega_{s}(K)\leq\Omega_{s}(K^{*}_{s})$ where
$K_{s}^{*}$ is a spherical ball such that
$\operatorname{area}(K_{s}^{*})=\operatorname{area}(K)$.

Unfortunately, our proof method does not characterize the equality case in
Theorem 3.8.

In analogy with the Euclidean case, the floating area $\Omega_{s}(K)$ is
closely related to how well a spherical convex body $K\subset\mathbb{S}^{n}$
can be approximated by spherical polytopes. The connection between the
floating area and the asymptotic best and random approximation of convex
bodies in spherical and hyperbolic spaces was studied by Besau, Ludwig and
Werner in [2]. The next result is an asymptotic formula for the best
approximation of spherical convex bodies in $\mathbb{S}^{n}$ by inscribed
spherical polytopes with at most $N$ vertices.

For $K\in\mathcal{K}_{p}(\mathbb{S}^{n})$, let
$\mathcal{P}_{N}^{\mathbb{S}^{n}}(K)$ denote the set of all spherical
polytopes inscribed in $K$ with at most $N$ vertices, and set

$\operatorname{dist}_{s}(K,\mathcal{P}_{N}^{\mathbb{S}^{n}}(K)):=\inf\\{\operatorname{vol}_{s}(K)-\operatorname{vol}_{s}(P):\,P\in\mathcal{P}_{N}^{\mathbb{S}^{n}}(K)\\}.$

By a compactness argument, there exists a best-approximating polytope
$\widehat{P}\in\mathcal{P}_{N}^{\mathbb{S}^{n}}(K)$ such that
$\operatorname{dist}_{s}(K,P_{N}^{\mathbb{S}^{n}}(K))=\operatorname{vol}_{s}(K)-\operatorname{vol}_{s}(\widehat{P})$.

######  Theorem 3.9.

For every $K\in\mathcal{K}(\mathbb{S}^{n})$ with $C^{2}_{+}$ boundary, we have

$\lim_{N\to\infty}\operatorname{dist}_{s}(K,\mathcal{P}^{\mathbb{S}^{n}}_{N}(K))N^{\frac{2}{n-1}}=\frac{1}{2}{\rm
del}_{n-1}\left[\Omega_{s}(K)\right]^{\frac{n+1}{n-1}}.$

###### Proof.

The proof is a suitable modification of the proof of [2, Theorem 2.4], and
thus, we only sketch it. First, we derive an asymptotic formula for the
weighted volume difference of any convex body $C\in\mathcal{K}^{n}$ with
$C^{2}_{+}$ boundary and a minimizing inscribed polytope with $N$ vertices as
$N\to\infty$. To do so, we can use a formula of Glasauer and Gruber [9] for
the weighted volume difference, combined with adaptations of the arguments of
Ludwig [14] from the general unconstrained case of weighted best approximation
to the inscribed case. As a result, we only need to replace ${\rm ldel}_{n-1}$
with ${\rm del}_{n-1}$ in [14, Theorem 3] to obtain the desired formula for
weighted inscribed approximation in the Euclidean case.

To obtain the desired formula for the inscribed approximation of spherical
convex bodies contained in an open hemisphere of class $C^{2}_{+}$, we can now
argue as Besau, Ludwig and Werner do in the proof of [2, Theorem 2.4], where
the more general case in which the polytopes are unconstrained was settled.
Simple modifications to their arguments yield the result, only now we use the
inscribed weighted approximation result, rather than the unconstrained one. ∎

Choosing $n=2$ in Theorem 3.9 and using ${\rm del}_{1}=1/6$, we obtain

(14)
$\lim_{N\to\infty}\operatorname{dist}_{s}(K,\mathcal{P}^{\mathbb{S}^{2}}_{N}(K))N^{2}=\frac{1}{12}\Omega_{s}(K)^{3}.$

We are now in position to prove Theorem 3.8.

###### Proof of Theorem 3.8.

Let $K\in\mathcal{K}_{p}(\mathbb{S}^{2})$ be centrally symmetric with
$C^{2}_{+}$ boundary. Combining (14) and Theorem 3.4, we get

$\displaystyle\frac{1}{12\operatorname{area}(K)}\Omega_{s}(K)^{3}$
$\displaystyle=\lim_{N\to\infty}\frac{\operatorname{dist}_{s}(K,\mathcal{P}_{N}^{\mathbb{S}^{2}}(K))}{\operatorname{area}(K)}N^{2}$
$\displaystyle\leq\lim_{N\to\infty}\frac{\operatorname{dist}_{s}(K_{s}^{*},\mathcal{P}_{N}^{\mathbb{S}^{2}}(K_{s}^{*}))}{\operatorname{area}(K_{s}^{*})}N^{2}=\frac{1}{12\operatorname{area}(K_{s}^{*})}\Omega_{s}(K_{s}^{*})^{3}.$

Since $\operatorname{area}(K)=\operatorname{area}(K_{s}^{*})$, the result
follows. ∎

## Acknowledgments

SH would like to thank Florian Besau for the valuable discussions on the topic
of the paper.

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