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# A spectral isoperimetric inequality on the $n$-sphere for the Robin-
Laplacian with negative boundary parameter

P. Acampora, A. Celentano, E. Cristoforoni, C. Nitsch, C. Trombetti

###### Abstract

For every given $\beta<0$, we study the problem of maximizing the first Robin
eigenvalue of the Laplacian $\lambda_{\beta}(\Omega)$ among convex (not
necessarily smooth) sets $\Omega\subset\mathbb{S}^{n}$ with fixed perimeter.
In particular, denoting by $\sigma_{n}$ the perimeter of the $n$-dimensional
hemisphere, we show that for fixed perimeters $P<\sigma_{n}$, geodesic balls
maximize the eigenvalue. Moreover, we prove a quantitative stability result
for this isoperimetric inequality in terms of volume difference between
$\Omega$ and the ball $D$ of the same perimeter.

Keywords: Robin Laplacian, negative boundary parameter, isoperimetric
inequalities for eigenvalues, curvature measures, convex sets

MSC 2020: 35P15, 58J50, 52A55

##  1 Introduction

Let $(M,g)$ be a complete Riemannian $n$-manifold, and let $\Omega\subset M$
be a bounded open set with smooth boundary. For every $\beta\in\mathbb{R}$
consider the Robin-Laplacian eigenvalue problem on $\Omega$, that is

$\begin{cases}-\Delta u=\lambda u&\text{in }\Omega,\\\\[5.0pt] \dfrac{\partial
u}{\partial\nu}+\beta u=0&\text{on }\partial\Omega,\end{cases}$ (1.1)

where $\Delta$ is the Laplace-Beltrami operator on $M$ and $\nu$ is the unit
outer normal to the boundary of $\Omega$. (1.1) admits an increasing sequence
of eigenvalues diverging to infinity. Moreover, if $\Omega$ is connected, any
first eigenfunction has a sign, so that, by linearity, the first eigenvalue
$\lambda_{\beta}(\Omega)$ is simple (see for example [27]).

Let $\lambda_{\beta}(\Omega)$ be the smallest eigenvalue for (1.1), then the
following variational characterization holds

$\lambda_{\beta}(\Omega)=\inf_{v\in
H^{1}(\Omega)}\dfrac{\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,d\mu+\beta\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\Omega}v^{2}\,d\mu}.$
(1.2)

Any minimizer $u$ of (1.2) is a weak solution to (1.1) for
$\lambda=\lambda_{\beta}(\Omega)$, that is

$\int_{\Omega}g(\nabla
u,\nabla\varphi)\,d\mu+\beta\int_{\partial\Omega}u\varphi\,d\mathcal{H}^{n-1}=\lambda_{\beta}(\Omega)\int_{\Omega}u\varphi\,d\mu,$

for every $\varphi\in H^{1}(\Omega)$. An immediate consequence of the
variational characterization (1.2) is the fact that the function

$\beta\in\mathbb{R}\mapsto\lambda_{\beta}(\Omega)\in\mathbb{R}$

is increasing. In particular, for $\beta=0$, the Robin boundary condition
coincides with the Neumann one and $\lambda_{0}(\Omega)=0$ with constant
eigenfunctions. Therefore, the first Robin eigenvalue is positive for
$\beta>0$ and negative for $\beta<0$.

Comparison theorems for the first Robin eigenvalue are widely studied in the
literature. The first example of such theorems is probably the one due to
Bossel in [9]: this result generalizes the Faber-Krahn inequality for the
first Robin eigenvalue with $\beta>0$ in the class of bounded open sets of the
Euclidean plane $\mathbb{R}^{2}$. Namely, let $\Omega\subset\mathbb{R}^{2}$ be
a bounded open set and let $B\subset\mathbb{R}^{2}$ be a ball having the same
area, then

$\lambda_{\beta}(\Omega)\geq\lambda_{\beta}(B).$ (1.3)

Daners generalized the previous result in [17] for bounded open subsets of the
Euclidean space $\mathbb{R}^{n}$.

In the context of Riemannian manifolds, one usually compares the Robin
eigenvalue of a bounded domain $\Omega$ in a complete manifold $M$ with the
one of a geodesic ball in an appropriate simply connected space form. In
particular, Chen, Cheng, and Li in [14] proved a Bossel-Daners inequality
(1.3) for bounded domains of a manifold $M$, where either $M$ is the
hyperbolic space or it is a compact manifold whose Ricci curvature tensor
satisfies a positive lower bound. As proved by Chen, li, and Wei in [15], the
inequality still holds in the case in which $M$ is a complete, non-compact,
manifold whose Ricci tensor is non-negative.

In the case $\beta<0$, Bareket in [8] famously conjectured that among all
Lipschitz sets of a given area in the Euclidian plane, the ball maximizes the
first Robin eigenvalue. Freitas and Krejčiřík in [20] disproved the
conjecture: they proved, via an asymptotic expansion, that, for
$\lvert\beta\rvert$ sufficiently large, the first Robin eigenvalue of an
annulus is larger than the one of the ball having the same measure. At the
same time, they proved that for smooth bounded subsets of the Euclidean plane,
the conjecture holds true provided that $\beta$ is sufficiently close to $0$.
However, fixing the perimeter leads to other interesting comparisons. Indeed,
Antunes, Freitas, and Krejčiřík in [5] proved a comparison theorem for the
first Robin eigenvalue, with $\beta<0$, under a perimeter constraint. Namely,
let $\Omega$ be a bounded open set with $C^{2}$ boundary in the Euclidean
plane and let $B\subset\mathbb{R}^{2}$ be a ball having the same perimeter,
then

$\lambda_{\beta}(\Omega)\leq\lambda_{\beta}(B).$ (1.4)

Bucur et al. in [11] proved that the inequality (1.4) holds true in any
dimension provided that we restrict the class of admissible sets to the one of
the convex sets, or, more in general, the inequality holds for any Lipschitz
set which can be written as $\Omega\setminus K$, where $\Omega$ is open and
convex and $K$ is a closed set in $\Omega$. Vikulova in [33] proved the result
in the Euclidean space $\mathbb{R}^{3}$ for bounded convex sets or connected
axiconvex sets whose boundary is diffeomorphic to the sphere.

In the context of Riemannian manifolds, Khalile and Lotoreichik in [24] proved
the following. Let $\Omega$ be a compact, two-dimensional, simply connected
Riemannian manifold with $C^{2}$ boundary and with Gauss curvature bounded
from above by a non-negative constant $\kappa_{0}$, and let $B$ be a geodesic
disc in the simply connected space form of Gauss curvature $\kappa_{0}$ with
the same perimeter as $\Omega$. Then, for every $\beta<0$, inequality (1.4)
holds.

Finally, in Riemannian manifolds, other comparison theorems for the first
Robin eigenvalue and domain monotonicity properties have been proved by Savo
in [31] and by Li and Wang in [27].

The main objective of this paper is to adapt the techniques of [11] to prove
the following theorem. Note that we denote by $\mathcal{H}^{n-1}$ the
Hausdorff measure, and we refer to Subsection 2.2 for the definition of strong
convexity.

######  Theorem 1.1.

Let $\Omega\subset\mathbb{S}^{n}$ be an open set such that $\bar{\Omega}$ is
strongly convex, and let $D$ be a strongly convex geodesic ball with

$\mathcal{H}^{n-1}(\partial\Omega)=\mathcal{H}^{n-1}(\partial D).$

Then

$\lambda_{\beta}(\Omega)\leq\lambda_{\beta}(D),$ (1.5)

and the equality holds if and only if, up to a translation, $\Omega=D$.

Notice that, thanks to [31, Theorem 5], we have that the eigenvalue is
increasing with respect to the inclusion among balls, so that (1.5) also holds
true replacing $D$ with $\mathbb{S}^{n}\setminus D$.

The proof relies on the use of the method of parallel coordinates (see [29]
and [16]) to construct a suitable test function on convex subsets of the
sphere $\mathbb{S}^{n}$. Nevertheless, the main difficulty here was to recover
classical results about convex sets on the sphere. In particular, the main
ingredients of the proof are:

  1. (i)

convexity properties of inner and outer parallel sets;

  2. (ii)

monotonicity of perimeters with respect to the inclusion for convex sets;

  3. (iii)

Steiner’s formulae;

  4. (iv)

Alexandrov-Fenchel inequality for the mean curvature.

To infer convexity properties of inner parallel and outer parallel sets (see
Subsection 2.1) we need some convexity property of the distance function
provided by Bangert in his paper [7] (see Theorem 2.6). The monotonicity of
the perimeter has been proved by Bangert in [6] (see Theorem 2.4). The
Steiner’s formulae have been extended to $C^{2}$ convex sets of the sphere by
Allendoerfer in [2] and the Alexandrov-Fenchel inequality has been recently
extended to $C^{2}$ convex sets of the sphere by Makowski and Scheuer in [28].
However, our result in Theorem 1.1 only requires the set to be convex: to
avoid the constraint on the regularity of the boundary, we recover a general
theory for Steiner’s formulae and curvature measures introduced by Federer in
[18] in $\mathbb{R}^{n}$ and successively generalized to simply connected
space forms by Kohlmann in [25]. We are then able to generalize Alexandrov-
Fenchel inequalities for general convex sets (see Subsection 2.3) by
approximating convex sets with smooth convex sets, using a result that has
been proved by Bangert in [7] (see also Theorem 2.7).

In addition, we are also able to adapt the techniques in [3] to prove a
stability result resumed in the following

######  Theorem 1.2.

Let $\Omega\subset\mathbb{S}^{n}$ be an open set such that $\overline{\Omega}$
is strongly convex, and let $D$ be a strongly convex geodesic ball such that

$\mathcal{H}^{n-1}(\partial\Omega)=\mathcal{H}^{n-1}(\partial D).$

For every $\beta<0$, let $u$ be an eigenfunction relative to
$\lambda_{\beta}(D)$, and let

$u_{m}=\min_{p\in\overline{D}}u(p).$

Then,

$\dfrac{\lambda_{\beta}(D)-\lambda_{\beta}(\Omega)}{\lvert\lambda_{\beta}(\Omega)\rvert}\geq\dfrac{u_{m}^{2}}{\lVert
u\rVert_{L^{2}(D)}^{2}}(\lvert D\rvert-\lvert\Omega\rvert).$ (1.6)

The paper is organized as follows. In Section 2 we give introductory notions
and classical tools of Riemannian manifolds and integration theory. In
Subsection 2.2 we give classical results and definitions about convexity in
Riemannian manifolds, with special attention to the convexity of the inner
parallel sets and convex approximation. In Subsection 2.3 we give the
definition of curvature measures, and we state the Steiner formula in
$\mathbb{S}^{n}$ and the Alexandrov-Fenchel inequality. In Section 3 we prove
Theorem 1.1 and Theorem 1.2. Finally, in Section 4 we discuss the limits of
the proof in the hyperbolic space.

##  2 Notation and tools

In the following, given a smooth, orientable Riemannian $n$-manifold $(M,g)$,
we will denote by $d$ the Riemannian distance

$d(p,q)=\min_{\begin{subarray}{c}\gamma\in C^{\infty}((0,1);M)\\\
\gamma(0)=p\\\
\gamma(1)=q\end{subarray}}\>\int_{0}^{1}g(\gamma^{\prime}(t),\gamma^{\prime}(t))\,dt$

induced by $g$; by $d\mu$ its volume form which is expressed locally in
coordinates as

$d\mu=\sqrt{\lvert\det(g_{ij})\rvert}dx_{1}\dots dx_{n};$

and we will denote by $\lvert\cdot\rvert$ the classical Riemannian volume

$\lvert E\rvert=\int_{E}\,d\mu.$

We let $TM$ denote the tangent bundle on $M$, by $\Gamma(TM)$ the sections of
the bundle, namely the space of vector fields, and by $T_{p}M$ the tangent
space at $p$. We also recall that for every $(v,p)\in TM$ a _geodesic_
starting from $p$ with velocity $v$ is the unique curve $\gamma=\gamma_{p,v}$
such that $\gamma(0)=p$, and $\gamma^{\prime}(0)=v$, and such that it solves
the system of equations written in local coordinates as (using the Einstein
notation on repeated indices)

$\gamma_{i}^{\prime\prime}(t)+\Gamma^{i}_{jk}(\gamma(t))\,\gamma^{\prime}_{j}(t)\,\gamma^{\prime}_{k}(t)=0,\qquad\qquad
i=1,\dots,n$

with $\Gamma^{i}_{jk}$ representing the Christoffel symbols of the metric $g$.
When $M$ is complete, we can extend the geodesics $\gamma_{p,v}\in
C^{\infty}(\mathbb{R};M)$, and we denote by

$\exp:TM\to M\qquad\exp_{p}:T_{p}M\to M$

the exponential map defined as

$\exp(p,v)=\exp_{p}(v)=\gamma_{p,v}(1).$ (2.1)

For every $p\in M$ we will denote the cutlocus of $p$ in $M$ as

$\operatorname{Cut}(p)=\exp_{p}(\partial\operatorname{seg}(p)),$

where

$\operatorname{seg}(p)=\Set{v\in T_{p}M}{\gamma_{p,v}\text{ minimizes the
distance $d(p,\gamma_{p,v}(1))$}}.$

We will denote by $\mathcal{H}^{k}$ the Hausdorff measure relative to the
Riemannian distance on $M$. When necessary, we will denote the Hausdorff
measure by $\mathcal{H}^{k}_{g}$ to highlight the dependence on the metric
$g$. We refer to [12, Section IV] for basic properties on this topic in the
Riemannian setting. We denote by $\sigma_{n}$ the $(n-1)$-dimensional measure
of the boundary of a hemisphere in the sphere $\mathbb{S}^{n}$ of sectional
curvature $1$. Moreover, we will denote by $\langle\cdot,\cdot\rangle$ the
canonical scalar product in $\mathbb{R}^{n}$.

###  2.1 General notions

In the following, we will need some approximation argument. Hence, we define
the Hausdorff distance of sets. Let us recall that given a closed set
$\Omega\subset M$ the distance from $\Omega$ is defined as

$d(p,\Omega)=\inf_{q\in\Omega}d(p,q).$

######  Definition .

Let $M$ be a Riemannian manifold, and let $K\subset M$ be a compact set. For
every $t\geq 0$, we define the _inner parallel set_

$(K)_{t}=\Set{p\in K}{d(p,\partial K)\geq t},$

and the _outer parallel_ set

$(K)^{t}=\Set{p\in M}{d(p,K)\leq t}.$

######  Definition  (Hausdorff distance).

Let $M$ be a Riemannian manifold, and let $K_{1},K_{2}\subset M$ be two
compact sets. We define the Hausdorff distance as

$d^{H}(K_{1},K_{2})=\inf\Set{t\geq
0}{\begin{gathered}K_{1}\subset(K_{2})^{t}\\\
K_{2}\subset(K_{1})^{t}\end{gathered}}$

We refer to [35, §2] for the following definitions.

######  Definition .

Let $M$ be a Riemannian $n$-manifold, and let $\Sigma\subset M$. We say that
$\Sigma$ is a _strongly Lipschitz submanifold_ of $M$ of dimension $k$ if for
every $p\in\Sigma$ there exist a $C^{1}$ chart $(U,\varphi)$ in $M$ around
$p$, an open set $U^{\prime}\subset\mathbb{R}^{k}$, and a Lipschitz function
$f:U^{\prime}\to\mathbb{R}^{n-k}$ such that

$\varphi(\Sigma\cap U)=\Set{(x,f(x))\in\varphi(U)}{x\in U^{\prime}}.$

######  Definition .

Let $M$ be a Riemannian $n$-manifold, and let $\Omega\subset M$. We say that
$\Omega$ _has strongly Lipschitz boundary_ if
$\Omega=\overline{\mathring{\Omega}}$, and $\partial\Omega$ is a strongly
Lipschitz submanifold of $M$ of dimension $n-1$.

######  Definition .

Let $X,Y$ be two metric spaces. We say that a homeomorphism

$f:X\to Y$

is _locally bi-Lipschitz_ if both $f$ and $f^{-1}$ are locally Lipschitz.

######  Definition .

Let $(M,g)$ be a Riemannian $n$-manifold, and let $\Sigma$ be a $C^{2}$
oriented, embedded $(n-1)$-submanifold of $M$. We define the _second
fundamental form_ $h$ of $\Sigma$ in $M$ as the 2-form such that for every
$X,Y\in\Gamma(T\Sigma)$

$h(X,Y)=g(X,\nabla_{Y}\nu),$

where $\nabla$ is the Levi-Civita connection of $M$, and $\nu$ is the normal
to $\Sigma$.

######  Proposition .

Let $M$ and $\Sigma$ as in Subsection 2.1. Then:

  1. (i)

$h$ is symmetric, namely

$h(X,Y)=h(Y,X)\qquad\forall X,Y\in\Gamma(T\Sigma);$

  2. (ii)

for every $\sigma\in\Sigma$ there exist $n-1$ eigenvalues
$k_{1}(\sigma)\leq\dots\leq k_{n-1}(\sigma)$ of $h$ and we say that $k_{i}$
are the _principal curvatures of $\Sigma$_.

######  Definition .

Let $M$ be a Riemannian $n$-manifold, let $\Sigma$ be a $C^{2}$ oriented,
compact, embedded $(n-1)$-submanifold of $M$. For every $p\in\Sigma$ and for
every $1\leq j\leq n-1$, we denote by

$H_{j}(p)=\sum_{1\leq i_{1}<\dots<i_{j}\leq n-1}k_{i_{1}}(p)\dots
k_{i_{j}}(p)$

the $j$-th homogeneous symmetric form of the principal curvatures, and

$H_{0}(p)=1.$

In particular, we say that $H_{1}(p)$ is the _mean curvature_ of $\Sigma$ in
$p$.

We now state the coarea and area formula.

######  Definition .

Let $V$ be a normed vector space of dimension $n$. For every $r=1,\dots,n$ we
denote by $\bigwedge\nolimits_{r}V$ the space of alternating $r$-forms on the
dual $V^{*}$.

If $V=T_{p}M$ is a tangent space for a Riemannian $n$-manifold $M$ at a point
$p$, for every $r\leq n$ we use the notation

$\bigwedge\nolimits_{r}M_{p}:=\bigwedge\nolimits_{r}T_{p}M$

to denote the inner product of $r$ copies of $T_{p}M$.

######  Definition .

Let $(M,g)$ be a Riemannian $n$-manifold of class $C^{1}$, let $(N,h)$ be a
Riemannian $k$-manifold of class $C^{1}$, let

$r=\min\\{n,k\\},$

and let $f:M\to N$ be a map such that $f$ is differentiable in $p\in M$. We
define the natural extension of $df_{p}$ to $\bigwedge_{r}M_{p}$ as the linear
map

$\wedge_{r}df_{p}:\bigwedge\nolimits_{r}M_{p}\to\bigwedge\nolimits_{r}N_{f(p)}$

such that for every $v_{1},\dots,v_{r}\in T_{p}M$

$\wedge_{r}df_{p}(v_{1}\wedge\dots\wedge v_{r})=df_{p}(v_{1})\wedge\dots
df_{p}(v_{r}).$

We define the _jacobian_ of $f$ as

$Jf(p)=\lVert\wedge_{r}df_{p}\rVert,$

where the norm $\lVert\cdot\rVert$ denotes the operatorial norm in the space
of linear applications $\mathcal{L}(\bigwedge_{r}M_{p},\bigwedge_{r}N_{f(p)})$
with the respective norms $\lVert\cdot\rVert_{g,p}$ and
$\lVert\cdot\rVert_{h,f(p)}$.

For the proof of the following theorem, we refer to [18, Theorem 3.1]

######  Theorem 2.1 (Coarea Formula).

Let $(M,g)$ be a Riemannian $n$-manifold, let $(N,h)$ be a Riemannian
$k$-manifold with $n\geq k$, and let $f:M\to N$ be a Lipschitz map. Then $f$
is $\mathcal{H}^{n}$-a.e. differentiable and for every
$\mathcal{H}^{n}$-integrable function $\varphi:M\to\mathbb{R}$ we have

$\int_{M}\varphi(x)\,J\\!f(x)\,d\mathcal{H}^{n}(x)=\int_{N}\int_{f^{-1}(y)}\varphi(z)\,d\mathcal{H}^{n-k}(z)\,d\mathcal{H}^{k}(y).$

For the following theorem we refer to [19, Theorem 3.2.5, Remark 3.2.46].

######  Theorem 2.2 (Area Formula).

Let $(M,g)$ be a Riemannian $n$-manifold, let $(N,h)$ be a Riemannian
$k$-manifold with $n\leq k$, and let $f:M\to N$ be a Lipschitz map. Then $f$
is $\mathcal{H}^{n}$-a.e. differentiable and for every
$\mathcal{H}^{n}$-measurable function $\varphi:M\to\mathbb{R}$ and we have

$\int_{M}\varphi(x)\,J\\!f(x)\,d\mathcal{H}^{n}(x)=\int_{N}\int_{f^{-1}(y)}\varphi(z)\,d\mathcal{H}^{0}(z)\,d\mathcal{H}^{k}(y).$

###  2.2 Convexity in Riemannian manifolds

In this section, we aim to give a general overview of convexity in Riemannian
manifolds, and then we will study properties of convex sets in the specific
case of the sphere $\mathbb{S}^{n}$. In order to give some convexity
definitions in the Riemannian setting, we introduce the notions of supporting
cone and normal cone. (We recall the definition of the exponential map in
(2.1).)

######  Definition .

Let $M$ be a Riemannian manifold and $C\subset M$ with non-empty interior. For
every $p\in\partial C$ we define the _(local) supporting cone_ of $C$ in $p$
as

$\mathcal{C}_{C}(p)=\Set{\xi\in
T_{p}M}{\exists\varepsilon>0:\exp_{p}(t\xi)\in\mathring{C}\quad\forall
t\in(0,\varepsilon)},$

and the _(internal) normal cone_ as its dual cone

$\mathcal{C}_{C}(p)^{*}=\Set{\nu\in T_{p}M}{\langle\nu,\xi\rangle\geq
0\quad\forall\xi\in\mathcal{C}_{C}(p)}.$

Then, recalling that we use the notation $\overline{pq}$ to denote the minimal
geodesic connecting $p$ and $q$ is unique in $M$, we give the following
definitions

######  Definition .

Let $M$ be a Riemannian manifold, and let $C_{1},C_{2}\subset M$. We say that:

  1. (a)

$C_{1}$ is _weakly convex_ if for every $p,q\in C_{1}$ there exists a minimal
geodesic $\gamma:[a,b]\to M$ connecting $p$ and $q$ contained in $C_{1}$;

  2. (b)

$C_{1}$ is _strongly convex_ if for every $p,q\in C_{1}$ there exists a unique
minimal geodesic $\overline{pq}$ connecting $p$ and $q$ in $M$, and
$\overline{pq}\subseteq C_{1}$;

  3. (c)

$C_{1}$ is _locally convex_ if for every $p\in\bar{C_{1}}$ there exists
$\varepsilon>0$ and a metric ball $B_{\varepsilon}(p)$ such that $C_{1}\cap
B_{\varepsilon}(p)$ is strongly convex;

  4. (d)

$C_{1}$ is _locally strictly convex_ if there exists a $\delta>0$ such that
for every point $p\in\partial C_{1}$ and for every
$\nu\in\mathcal{C_{1}}_{C}(p)^{*}$ the following holds: there exists an
hypersurface $H$ orthogonal to $\nu$ in $p$ such that $H\cap C_{1}=\\{p\\}$
and its second fundamental form in $p$ with respect to $\nu$ has eigenvalues
greater than $\delta$;

  5. (e)

$C_{1}$ is _totally convex_ in $C_{2}$ if $C_{1}\subseteq\mathring{C_{2}}$ and
for every $p,q\in C_{1}$ and every geodesic

$\gamma:[a,b]\to C_{2}$

connecting $p$ and $q$ inside $C_{2}$ we have $\gamma([a,b])\subseteq C_{1}$.

We refer to [13] for definitions (a)-(c), to [7] for definition (d), and to
[6] for definition (e).

We now give some useful properties about convex sets in the sphere.

######  Remark .

Recall that the definition of strong convexity is actually imposing some
geometric constraint on the set $C$. For instance, on the sphere
$\mathbb{S}^{n}$ we have that if $C\subseteq\mathbb{S}^{n}$ is a closed
strongly convex set, then $C$ is contained in an open hemisphere. Indeed, let
$C\subseteq\mathbb{S}^{n}$ be a closed strongly convex set. By definition of
strong convexity, we have that if $p\in C$ then necessarily the antipodal
point $-p\notin C$. Therefore, we can find a plane separating $C$ and its
antipodal set $-C$: indeed,

$\Omega^{+}:=\Set{tx\in\mathbb{R}^{n+1}}{\begin{gathered}t>0,\\\ x\in
C\end{gathered}}$

and

$\Omega^{-}:=\Set{tx\in\mathbb{R}^{n+1}}{\begin{gathered}t>0,\\\
x\in-C\end{gathered}}$

are two disjoint convex cones in $\mathbb{R}^{n+1}$, and they can be separated
by a plane passing through the origin. This in particular implies that $C$ is
contained in a hemisphere.

######  Remark .

If $C\subset\mathbb{S}^{n}$ is weakly convex and it is contained in a
hemisphere, then it is strongly convex, since for every couple of points
$p,q\in C$ there exists a unique minimal geodesic connecting them.

######  Remark .

Notice that if $C_{1},C_{2}\subset\mathbb{S}^{n}$ are two strongly convex sets
such that $C_{1}\subseteq C_{2}$, then $C_{1}$ is totally convex in $C_{2}$.
Indeed, since $C_{2}$ is contained in a hemisphere, then for every couple of
points $p,q\in C_{1}$, the unique minimal geodesic $\overline{pq}$ connecting
$p$ and $q$ is also the unique geodesic connecting $p$ and $q$ contained
$C_{2}$.

Notice that the definition of totally convex set becomes trivial when $M$ is a
compact manifold and we take $C_{2}=M$. See for instance [6, Corollary 1] for
the following

######  Proposition .

Let $M$ be a compact connected Riemannian manifold, and let $C\subseteq M$ be
a totally convex set in $M$. Then $C=M$.

######  Remark .

Notice that if $C$ is strongly convex, then it is connected and locally
convex.

Notice also that if $C_{1}$ is strongly convex and $C_{1}\subset C_{2}$ is
totally convex in $C_{2}$, then $C_{1}$ is strongly convex.

In $\mathbb{S}^{n}$, open, connected, locally convex sets contained in a
hemisphere have to be strongly convex. Indeed, we can characterize weak
convexity with some geometric properties of the boundary. Let us introduce the
notion of supporting element (see [13, 1]).

######  Definition .

Let $M$ be a Riemannian manifold, and let $C\subseteq M$ be an open set. Let
$p\in\partial C$, and for some $\nu\in T_{p}M$ define

$H_{p}=\Set{\xi\in T_{p}M}{\langle\nu,\xi\rangle<0}.$

We say that:

  1. (i)

the half-space $H_{p}$ is a _supporting element_ for $C$ in $p$ if for every
$q\in\mathring{C}$ and for every minimal geodesic

$\gamma:[0,1]\to M$

such that $\gamma(0)=p$ and $\gamma(1)=q$, we have $\gamma^{\prime}(0)\in
H_{p}$;

  2. (ii)

the half-space $H_{p}$ is a _locally supporting element_ for $C$ in $p$ if
there exists a neighbourhood $U$ of $p$ such that $H_{p}$ is a supporting
element for $U\cap C$ in $p$.

Let $M$ be a Riemannian manifold, and for every $p\in M$, let
$\operatorname{Cut}(p)$ be the cut-locus of $p$. We refer to [1, Proposition
2] for the following result.

######  Proposition .

Let $M$ be a Riemannian manifold, and let $C\subset M$ be connected and open.
The set $C$ is weakly convex if and only if for every point $p\in\partial C$
there exists a locally supporting element and
$C\setminus\operatorname{Cut}(p)$ is connected.

We also have that a locally supporting element always exists for open, locally
convex sets. Indeed, Cheeger and Gromoll in [13, Theorem 1.6, Lemma 1.7]
proved a result summarized in Theorem 2.3 (see also the comments between Lemma
1.7 and Proposition 1.8); notice that Cheeger and Gromoll work with closed
sets, but if $C$ is locally convex, then also $\bar{C}$ is a locally convex
set, and $\partial C=\partial\bar{C}$. On the other hand, by definition, a
supporting element for $\bar{C}$ is also a supporting element for $C$.

######  Theorem 2.3.

Let $M$ be a Riemannian manifold of dimension $n$, and let $C\subseteq M$ be a
non-empty, open, locally convex set. Then $\partial C$ is an embedded
$(n-1)$-dimensional topological submanifold of $M$, and it has a supporting
element in every point $p\in\partial C$.

Joining Subsection 2.2 and Theorem 2.3, we get on the sphere $\mathbb{S}^{n}$
the following.

######  Proposition .

Let $C\subset\mathbb{S}^{n}$ be a closed, connected, locally convex set
contained in an open hemisphere. Then $C$ is strongly convex.

###### Proof.

The local convexity of $C$ and the fact that it is connected ensure that
$\mathring{C}$ is connected (see for instance [13, Lemma 1.5]). Therefore, we
may apply Theorem 2.3 to $\mathring{C}$, so that every point
$p\in\partial\mathring{C}$ admits a supporting element. Moreover, since $C$ is
contained in a hemisphere, we also have that

$\mathring{C}\setminus\operatorname{Cut}(p)=\mathring{C},$

which is connected. Therefore, we can apply Subsection 2.2, and get that
$\mathring{C}$ is weakly convex, and, in particular, as in Subsection 2.2,
strongly convex. Finally, observing that closedness and local convexity ensure
$C=\bar{\mathring{C}}$ (see [13, Theorem 1.6]), then $C$ inherits the strong
convexity of $\mathring{C}$.

∎

The following theorem is due to Bangert in [6, Theorem 1].

######  Theorem 2.4 (Monotonicity of perimeter).

Let $M$ be a Riemannian manifold, and let $C_{1},C_{2}\subseteq M$ such that
$C_{1}$ is totally convex in $C_{2}$, and $\mathring{C_{1}}\neq\emptyset$.
Assume moreover that $C_{2}$ has strongly Lipschitz boundary, and $\lvert
C_{2}\setminus C_{1}\rvert<+\infty$. Then

$\mathcal{H}^{n-1}(\partial C_{1})\leq\mathcal{H}^{n-1}(\partial C_{2}).$

The proof of this theorem in the Euclidean case only relies on proving that
the projection onto the convex set $C_{1}$ is a $1$-Lipschitz function (see
for instance [10, Proposition 5.3]), while the Riemannian case requires a
different proof. Even if the monotonicity theorem requires some regularity on
the external set $C_{2}$, we can still prove that this is not restrictive in
the case in which $C_{2}$ is locally convex. Indeed, we have the following
result due to Walter in [35, Theorem 6.1].

######  Theorem 2.5.

Let $M$ be a Riemannian manifold, and let $C\subset M$ be a closed, locally
convex set. Then $C$ has strongly Lipschitz boundary.

Joining Theorem 2.4 and Theorem 2.5, we get:

######  Corollary .

Let $C_{1},C_{2}\subseteq\mathbb{S}^{n}$ be two closed strongly convex sets
such that $\mathring{C_{1}}\neq\emptyset$. If $C_{1}\subseteq C_{2}$, then

$\mathcal{H}^{n-1}(\partial C_{1})\leq\mathcal{H}^{n-1}(\partial C_{2}).$

###### Proof.

It is sufficient to notice that, as in Subsection 2.2, $C_{1}$ is totally
convex in $C_{2}$. Indeed, Theorem 2.5 ensures the strongly Lipschitz
regularity of the boundary, and Theorem 2.4 applies. ∎

We now give some definitions of convexity of continuous functions on
Riemannian manifolds, see for instance [21, §1] for a reference on the topic.

######  Definition .

Let $M$ be a Riemannian manifold, and let $f:M\to\mathbb{R}$ be a continuous
function. We say that:

  1. (a)

$f$ is _convex_ if for every geodesic $\gamma:[a,b]\to M$ we have
$f\circ\gamma$ is convex on $[a,b]$;

  2. (b)

$f$ is _strictly convex_ if for every $p\in M$ and for every convex function
$\varphi\in C^{\infty}(M)$ there exists an $\varepsilon>0$ such that
$f-\varepsilon\varphi$ is convex in a small neighbourhood of $p$.

These definitions are related to the geometry of the sublevel sets.

######  Proposition .

Let $M$ be a Riemannian manifold, and let $f:M\to\mathbb{R}$ be a continuous
function. Then:

  1. (i)

if $f$ is convex, then for every $t\in\mathbb{R}$ the set $\Set{x\in
M}{f(x)<t}$ is totally convex in $M$;

  2. (ii)

assume that $f$ is strictly convex, and $M$ is weakly convex; for every
$t\in\mathbb{R}$, if the set $\Set{x\in M}{f(x)<t}$ is compact, then it is
locally strictly convex.

###### Proof.

We only show (i), and we refer to [7, Lemma 2.4] for the proof of (ii) (note
that the assumption on the weak convexity of $M$ ensures the connectedness of
the sublevel set of $f$). Let $\gamma:[a,b]\to M$ be a geodesic, and assume
that

$f(\gamma(a))<t\qquad\qquad f(\gamma(b))<t.$

Then, by the definition of convexity, for every $\alpha\in[0,1]$,

$f\bigl{(}\gamma\bigl{(}a+\alpha(b-a)\bigr{)}\bigr{)}\leq(1-\alpha)f\bigl{(}\gamma(a)\bigr{)}+\alpha
f\bigl{(}\gamma(b)\bigr{)}<t,$

and the assertion is proved. ∎

We aim to inspect the geometric properties of inner parallel and outer
parallel of convex sets. Cheeger and Gromoll, in [13, Theorem 1.10] proved
that, for a given convex set $C$ in a Riemannian manifold $M$ with positive
sectional curvatures, the distance function

$\rho(x)=-d(x,\partial C)$

is convex in $\mathring{C}$. This implies that the inner parallel sets $C_{t}$
are totally convex in $\mathring{C}$. However, we will need some more refined
results that can be found in [7, Theorem 2.1, Theorem 2.3], and we summarize
in the following. Let $C\subset M$, and denote by $\rho=\rho_{C}$ the signed
distance function

$\rho(x)=\begin{cases}-d(x,\partial C)&\text{if }x\in\mathring{C},\\\
d(x,C)&\text{if }x\notin\mathring{C}.\end{cases}$

Then we have

######  Theorem 2.6.

Let $M$ be a Riemannian manifold, and let $C$ be a connected, compact, locally
convex set. Then the following hold:

  1. (i)

if $C$ is locally strictly convex, then there exists $\delta>0$ such that the
function

$\rho+\frac{1}{2}\rho^{2}$

is strictly convex on $\mathring{C}^{\delta}\setminus C$;

  2. (ii)

if the sectional curvatures on $C$ are negative, then there exists $\delta>0$
such that the function

$\rho+\frac{1}{2}\rho^{2}$

is strictly convex on $\mathring{C}^{\delta}\setminus C$;

  3. (iii)

if the sectional curvatures on $C$ are positive, then the function

$\rho-\log(-\rho)$

is strictly convex on $\mathring{C}$.

######  Remark .

Despite [7, Theorem 2.1] only proves (i), result (ii) directly follows from
the same proof using a negative upper bound on the sectional curvatures to
conclude (see also the proof of [7, Corollary 2.6]).

######  Corollary .

Let $C\subset\mathbb{S}^{n}$ be a closed strongly convex set. Then:

  1. (i)

if $C$ is strongly convex and locally strictly convex, then for small
$\delta>0$ we have that the outer parallel sets $(C)^{t}$ are strongly convex
and locally strictly convex for every $t<\delta$;

  2. (ii)

the inner parallel sets $(C)_{t}$ are locally strictly convex and strongly
convex for every $t>0$.

###### Proof.

By the condition (i) in Theorem 2.6 we get that $(C)^{t}$, for small values of
$t$ is locally convex. Indeed, for every interior point $p$ of $(C)^{t}$ it is
sufficient to observe that a small strongly convex ball contained in $(C)^{t}$
always exists. If $p\in\partial(C)^{t}$, since we can find a small strongly
convex ball $B$ contained in $(\mathring{C})^{\delta}\setminus C$, then the
convexity of the function $\rho+\rho^{2}/2$ ensures that $B\cap(C)^{t}$ is
strongly convex.

Moreover, $C$ is connected and contained in a hemisphere, as already seen in
Subsection 2.2. Therefore, $(C)^{t}$, for small $t$, is connected and
contained in the same hemisphere, which implies, by Subsection 2.2, that
$(C)^{t}$ is strongly convex. Finally, by Subsection 2.2, we also get that for
small $t$ the set $(C)^{t}$ is locally strictly convex.

Let us now study the inner parallels $(C)_{t}$. Analogously to the case of the
outer parallels, condition (iii) in Theorem 2.6 yields that the inner parallel
sets $(C)_{t}$ are locally strictly convex. Moreover, the convexity of the
function $\rho-\log(-\rho)$ ensures that the sets $(C)_{t}$ are totally convex
in $\mathring{C}$ (see (i) in Subsection 2.2). Since $C$ is strongly convex,
then the total convexity of $(C)_{t}$ in $\mathring{C}$ gives that the inner
parallel sets $(C)_{t}$ are strongly convex. ∎

Now we state an approximation theorem proved by Bangert in [7, Theorem 2.2,
Corollary 2.5, Corollary 2.6].

######  Theorem 2.7.

Let $M$ be a Riemannian manifold, and let $C\subset M$ be a connected,
compact, locally convex set such that $\mathring{C}\neq\emptyset$. Moreover,
assume that either:

  1. (a)

$C$ is locally strictly convex;

  2. (b)

the sectional curvatures are positive on $C$;

  3. (c)

the sectional curvatures are negative on $C$;

then there exists a sequence of connected, compact, locally convex sets
$C_{k}$ with $C^{\infty}$ boundaries such that

$\lim_{k\to+\infty}d^{H}(C_{k},C)+d^{H}(\partial C_{k},\partial C)=0.$

######  Remark .

Results (b) and (c) are a direct consequence of (a) and Theorem 2.6 joint with
Subsection 2.2: in the case (b) one approximates the inner parallel sets,
while in the case (c) one approximates the outer parallel sets.

######  Corollary .

Let $C\subset\mathbb{S}^{n}$ be a closed strongly convex set such that
$\mathring{C}\neq\emptyset$. Then there exists a sequence of closed strongly
convex sets $C_{k}$ with $C^{\infty}$ boundaries such that

$\lim_{k\to+\infty}d^{H}(C_{k},C)+d^{H}(\partial C_{k},\partial C)=0.$

###### Proof.

Since the sectional curvatures in $\mathbb{S}^{n}$ are positive, Theorem 2.7
applies and we find an approximating sequence of connected, compact, locally
convex sets $C_{k}$ with $C^{\infty}$ boundaries and such that
$\mathring{C}_{k}\neq\emptyset$. Therefore, the Hausdorff convergence also
allows us to assume that $C_{k}$ are contained in the same hemisphere in which
$C$ is contained. By Subsection 2.2, we get that $C_{k}$ are strongly convex.
∎

###  2.3 Curvature measures

In this section we define curvature measures introduced in $\mathbb{R}^{n}$ by
Federer in [18] and explicitly computed by Zähle in [37], while successively
extended to simply connected space forms by Kohlmann in [25].

As a first step, we define sets of positive reach. Given a Riemannian manifold
$M$, for every $p\in M$ and for every $r>0$ we denote by $B_{r}(p)$ the metric
ball centered in $p$ of radius $r>0$. For small enough $r>0$ we have that
$B_{r}(p)$ coincides with the geodesic ball $\exp_{p}(B_{r}(0))$.

######  Definition .

Let $M$ be a Riemannian manifold, and let $\Omega\subset M$ be a non-empty
set. For every $p\in M$ we call _metric projection_ of $p$ onto $\Omega$ any
$q\in\bar{\Omega}$ such that

$d(p,q)=d(p,\Omega).$

When it is unique we write $q=\sigma_{\Omega}(p)$.

######  Definition .

Let $M$ be a Riemannian manifold, and let $\Omega\subset M$ be a non-empty
set. For every $q\in\Omega$ we define the _reach_ of $q$ with respect to
$\Omega$ as

$\mathcal{R}(q)=\sup\Set{r>0}{\forall p\in B_{r}(q)\text{ there exists a
unique metric projection of $p$ onto $\Omega$}};$

we define the _reach_ of $\Omega$ as

$\mathcal{R}(\Omega)=\inf_{q\in\Omega}\mathcal{R}(q);$

we say that $\Omega$ is _of positive reach_ if $\mathcal{R}(\Omega)>0$.

In $\mathbb{R}^{n}$ we have that every convex set $C$ is a set of positive
reach with

$\mathcal{R}(C)=+\infty.$

On simply connected space forms similar but slightly different results hold
for connected, compact, locally convex sets. First, we state a result due to
Walter in [34, Theorem 1].

######  Proposition .

Let $M$ be a Riemannian manifold, and let $C\subseteq M$ be a closed locally
convex set. Then $C$ is of positive reach.

In the specific case of simply connected space forms we have the following
result that can be found in [26, Lemma 2.2].

######  Proposition .

Let $M$ be a simply connected space form of curvature $\kappa$, and let
$C\subseteq M$ be a connected, compact, locally convex set. Then:

  1. (i)

if $\kappa<0$, then

$\mathcal{R}(C)=+\infty;$

  2. (ii)

if $\kappa>0$, then

$\mathcal{R}(C)\geq\frac{\pi}{2}\frac{1}{\sqrt{\kappa}}.$

######  Definition .

Let $(M,g)$ be a Riemannian manifold, let $\Omega\subseteq M$ be a set of
positive reach, and let $U$ be an open neighborhood of $\Omega$ such that the
metric projection $\sigma_{\Omega}$ is well defined for every $p\in U$. We
denote by

$\nu_{\Omega}(p)=\frac{\exp^{-1}_{\sigma_{\Omega}(p)}(p)}{\lVert\exp^{-1}_{\sigma_{\Omega}(p)}(p)\rVert_{g}}\in
TM,$

and we define the _unit normal bundle_

$\mathcal{N}(\Omega)=\nu_{\Omega}(U\setminus\Omega).$

Since we are going to integrate over $\mathcal{N}(\Omega)$, we need some
regularity property of the normal bundle, that is proved in [35, Theorem 4.3].
In the following, we are equipping $TM$ with the canonical Sasaki metric.

######  Theorem 2.8.

Let $M$ be a Riemannian $n$-manifold, and let $\Omega\subseteq M$ be a set of
positive reach. Then $\mathcal{N}(\Omega)$ is a strongly Lipschitz
$(n-1)$-submanifold of $TM$. Moreover, if $\Omega$ is compact, then
$\mathcal{N}(\Omega)$ is compact and there exists $\eta=\eta(\Omega)>0$ such
that for every $0<r<\eta$, we have that

${\left.\kern-1.2pt\nu_{\Omega}\vphantom{\big{|}}\right|_{\partial(\Omega)^{r}}}:\partial(\Omega)^{r}\to\mathcal{N}(\Omega)$

is a locally bi-Lipschitz homeomorphism.

######  Definition .

Let $M$ be a Riemannian $n$-manifold, let $\Sigma$ be a $C^{2}$ oriented,
compact, embedded $(n-1)$-submanifold of $M$ of positive reach, and let
$v\in\mathcal{N}(\Sigma)$. Let us denote by

$\Pi:TM\to M$

the canonical projection such that $\Pi(p,\xi)=p$ for every $(p,\xi)\in TM$.
For every $1\leq j\leq n-1$ we denote by $k_{j}(v):=k_{j}(\Pi(v))$ the
principal curvatures of $\Sigma$ in $\Pi(v)$, and by

$H_{j}(v)=\sum_{1\leq i_{1}<\dots<i_{j}\leq n-1}k_{i_{1}}(v)\dots
k_{i_{j}}(v)$

the $j$-th homogeneous symmetric form of the principal curvatures. We also
denote by

$H_{0}(v)=1.$

######  Definition .

Let $M$ be a complete Riemannian $n$-manifold of constant sectional curvature
$\kappa$. We define the functions

$\operatorname{sn}_{\kappa}(t)=\begin{dcases}\frac{1}{\sqrt{-\kappa}}\sinh(\sqrt{-\kappa}t)&\text{if
}\kappa<0,\\\ t&\text{if }\kappa=0\\\
\frac{1}{\sqrt{\kappa}}\sin(\sqrt{\kappa}t)&\text{if }\kappa>0,\\\
\end{dcases}$

and $\operatorname{cn}_{\kappa}$=$\operatorname{sn}_{\kappa}^{\prime}$. We
also let for $1\leq j\leq n$

$L_{j}(t):=\int_{0}^{t}\operatorname{cn}_{\kappa}^{n-j}(t)\operatorname{sn}_{\kappa}^{j-1}(t)\,dt,$

and

$L_{0}(t)=1.$

The following theorem is due to Kohlmann in [25], but we also point out that
Allendoerfer in [2] proves a Steiner formula for regular convex sets in
spheres with different techniques.

######  Theorem 2.9 (Steiner formula on simply connected space forms).

Let $M$ be a simply connected space form of dimension $n$ and curvature
$\kappa$, and let $\Omega\subset M$ be a set of positive reach. Let $U$ be an
open set in which the metric projection $\sigma_{\Omega}$ is well defined. For
every $j=0,\dots,n$ there exist Radon measures $\Phi_{j}(\Omega\,;\cdot)$ on
$U$ such that the following hold: if $E\subset M$ is a bounded Borel set, and
$s>0$ is such that

$\sigma_{\Omega}^{-1}(E)\cap\overline{(\Omega)^{s}}\subset U,$

then we have

$\mathcal{H}^{n-1}(\sigma_{\Omega}^{-1}(E)\cap\partial(\Omega)^{s})=\sum_{r=0}^{n-1}\operatorname{cn}_{\kappa}^{r}(s)\operatorname{sn}_{\kappa}^{n-1-r}(s)\Phi_{r}(\Omega;E),$

and

$\lvert\sigma_{\Omega}^{-1}(E)\cap(\Omega)^{s}\rvert=\sum_{r=0}^{n}L_{n-r}(s)\Phi_{r}(\Omega;E).$

In particular,

$\Phi_{n}(\Omega\,;E)=\lvert\Omega\cap E\rvert$

Moreover, if $\partial\Omega$ is a $C^{2}$ compact, embedded
$(n-1)$-submanifold of $M$, then for every bounded Borel set $E\subset M$ and
for every $r=0,\dots,n-1$ we have

$\Phi_{r}(\Omega;E)=\int_{\partial\Omega\cap
E}H_{n-1-r}(p)\,d\mathcal{H}^{n-1}(p).$ (2.2)

In the following, we denote by $\Phi_{r}(\Omega)$ the total measure, namely

$\Phi_{r}(\Omega)=\Phi_{r}(\Omega;M).$

######  Remark .

Notice that for sets $\Omega$ such that $\partial\Omega$ is of class $C^{2}$
the definition of the _curvature measures_ $\Phi_{r}$ given in [25, Theorem
2.7] is equivalent to (2.2). Indeed, let $\Omega$ be such that
$\partial\Omega$ is a $C^{2}$ compact, embedded $(n-1)$-submanifold of $M$.
Using the explicit definition of the measures $\Phi_{r}$ in [25], we have

$\Phi_{r}(\Omega;E)=\int_{\mathcal{N}(\Omega)\cap\Pi^{-1}(E)}H_{n-1-r}(v)\prod_{i=1}^{n-1}\frac{1}{\sqrt{1+k_{i}(v)^{2}}}d\mathcal{H}^{n-1}(v),$

where $H_{j}(v)$ and $k_{i}(v)$ are defined in Subsection 2.3. We start by
noticing that the regularity on $\partial\Omega$ ensures that

${\left.\kern-1.2pt\nu_{\Omega}\vphantom{\big{|}}\right|_{\partial\Omega}}:\partial\Omega\to
TM$

is a $C^{1}$ map, and using Area Formula (Theorem 2.2) with the change of
variables $\nu_{\Omega}(p)=v$, we have

$\Phi_{r}(\Omega;E)=\int_{\partial\Omega\cap
E}H_{n-1-r}(p)\prod_{i=1}^{n-1}\frac{1}{\sqrt{1+k_{i}(p)^{2}}}\,J\nu_{\Omega}\,d\mathcal{H}^{n-1}.$

In particular, as Kohlmann computed in [25, Equation (2.6) for
$\varepsilon=0$] (to help the reader compare the following equation with
Kohlmann’s, we recall that: $j_{1}=\operatorname{sn}_{\kappa}$ and
$j_{2}=\operatorname{cn}_{\kappa}$, while the functions $f_{i}$ are defined in
[25, Equation (1.20)]),

$J\nu_{\Omega}=\prod_{i=1}^{n-1}\sqrt{1+k_{i}(p)^{2}},$

and we have (2.2).

######  Remark .

Let $g$ denote the metric on the simply connected space form $M$. In the
following, we explicit the dependence on the metric. Notice that if $\Omega$
is a set of positive reach with $\partial\Omega$ strongly Lipschitz, then we
have that for every open set $E$

$\Phi_{n-1}^{g}(\Omega;E)=\mathcal{H}^{n-1}_{g}(\partial\Omega\cap E).$ (2.3)

Indeed, since the Steiner formula holds, then for every open set $E$ we have

$\Phi_{n-1}^{g}(\Omega;E)=\lim_{s\to
0^{+}}\frac{\lvert(\Omega^{s}\setminus\Omega)\cap E\rvert_{g}}{s},$

which is the definition of (relative) Minkowski perimeter (or
$(n-1)$-dimensional Minkowski content). If $M=\mathbb{R}^{n}$ equipped with
the Euclidean metric, then the equality (2.3) is classical (see for instance
[4, Theorem 2.106]). If $M$ is a generic simply connected space form, then it
is possible to obtain (2.3) from the Euclidean case using normal coordinates.
Let $p_{0}\in\partial\Omega$ and let $\varepsilon>0$, there exists a
$\delta=\delta(p_{0},\varepsilon)>0$ such that we can define an exponential
normal chart mapping onto
$\mathcal{U}_{\varepsilon}=\exp_{p_{0}}^{-1}(B_{\delta}(p_{0}))$ such that the
metric $g$ in coordinates is given by $g_{ij}=\delta_{ij}+O(\varepsilon)$. In
particular, if $g_{e}$ denotes the Euclidean metric on $\mathbb{R}^{n}$, the
diffeomorphism

$\operatorname{Id}:(\mathcal{U}_{\varepsilon},g)\to(\mathcal{U}_{\varepsilon},g_{e})$

is a bi-Lipschitz function with

$\operatorname{Lip}(\operatorname{Id})\leq
1+\varepsilon\qquad\operatorname{Lip}(\operatorname{Id}^{-1})\leq
1+\varepsilon.$

Therefore, if we denote by $\mathcal{H}^{n-1}_{g}$ the $(n-1)$-Hausdorff
measure with respect to the metric $g$, by $\lvert\cdot\rvert_{g}$ the
Riemannian volume, by $\mathcal{H}^{n-1}_{e}$ the $(n-1)$-Hausdorff measure
with respect to the Euclidean metric $g_{e}$, and by $\lvert\cdot\rvert_{e}$
the Lebesgue measure on $\mathbb{R}^{n}$, then we get for every Borel set
$A\subset\mathcal{U}_{\varepsilon}$ the following estimates (up to changing
$\varepsilon$)

$\begin{gathered}(1+\varepsilon)^{-1}\mathcal{H}^{n-1}_{g}(A)\leq\mathcal{H}^{n-1}_{e}(A)\leq(1+\varepsilon)\mathcal{H}^{n-1}_{g}(A),\\\\[9.0pt]
(1+\varepsilon)^{-1}\lvert A\rvert_{g}\leq\lvert
A\rvert_{e}\leq(1+\varepsilon)\lvert A\rvert_{g},\\\\[9.0pt]
\Set{x\in\mathcal{U}_{\varepsilon}}{d_{g}(x,A)<(1+\varepsilon)^{-1}}\subset\Set{x\in\mathcal{U}_{\varepsilon}}{d_{e}(x,A)<s}\subset\Set{x\in\mathcal{U}_{\varepsilon}}{d_{g}(x,A)<(1+\varepsilon)s}.\end{gathered}$
(2.4)

Using the estimates (2.4) and the fact that the equality (2.3) holds on
$\mathbb{R}^{n}$ for $\mathcal{H}^{n-1}_{e}$, then we get (up to choosing a
smaller $\varepsilon$) for every $r<\delta$,

$(1-\varepsilon)\mathcal{H}^{n-1}_{g}(\partial\Omega\cap
B_{r}(p_{0}))\leq\Phi_{n-1}^{g}(\Omega;B_{r}(p_{0}))\leq(1+\varepsilon)\mathcal{H}^{n-1}_{g}(\partial\Omega\cap
B_{r}(p_{0})),$ (2.5)

where $B_{r}(p_{0})$ denotes the Euclidean ball of radius $r$ centered in
$p_{0}$. Equation (2.5) in particular implies that the measure
$\Phi_{n-1}(\Omega;\cdot)$ is absolutely continuous with respect to
${\left.\kern-1.2pt\mathcal{H}^{n-1}_{g}\vphantom{\big{|}}\right|_{\partial\Omega}}$,
and that there exists a $\mathcal{H}^{n-1}_{g}$-measurable density $\rho$ such
that

$\Phi_{n-1}(\Omega;E)=\int_{\partial\Omega\cap E}\rho\,d\mathcal{H}^{n-1}_{g}$

with

$1-\varepsilon\leq\rho\leq 1+\varepsilon.$

Sending $\varepsilon$ to 0 we get $\rho=1$ and (2.3).

######  Remark .

Notice that in the case $\kappa=0$ we get the Steiner polynomial

$\lvert\sigma_{\Omega}^{-1}(E)\cap(\Omega)^{s}\rvert=\lvert\Omega\cap
E\rvert+\sum_{k=1}^{n}\frac{s^{k}}{k}\Phi_{n-k}(\Omega;E).$

We now give a continuity property for the curvature measures, and we refer to
[26, Theorem 2.4] for the proof.

######  Theorem 2.10.

Let $M$ be a simply connected space form of dimension $n$ and curvature
$\kappa$. Let $\Omega_{k}\subset M$ be a sequence of compact sets with non-
empty boundaries. Let us assume that for some $\delta>0$ and for some compact
set $\Omega\subset M$ we have

$\mathcal{R}(\Omega_{k})\geq\delta\qquad\qquad\lim_{k}d^{H}(\Omega_{k},\Omega)=0.$

Then for every $r=0,\dots,n$

$\Phi_{r}(\Omega_{k};\cdot)\xrightharpoonup{\hskip
15.0pt}\Phi_{r}(\Omega;\cdot)$

in the sense of Radon measures.

Finally, we state an Alexandrov-Fenchel inequality on the sphere comparing the
curvature measure $\Phi_{n-2}$ with $\Phi_{n-1}$, and we refer to [28, Theorem
1.5] for the proof of the regular case.

######  Theorem 2.11.

Let $\Omega\subset\mathbb{S}^{n}$ be a closed strongly convex set with $C^{2}$
boundary. Then

$\left(\frac{\Phi_{n-2}(\Omega)}{(n-1)\sigma_{n}}\right)^{2}\geq\left(\frac{\Phi_{n-1}(\Omega)}{\sigma_{n}}\right)^{\frac{2(n-2)}{n-1}}-\left(\frac{\Phi_{n-1}(\Omega)}{\sigma_{n}}\right)^{2},$

and the equality holds if and only if $\Omega$ is a geodesic ball.

######  Corollary .

Let $\Omega\subset\mathbb{S}^{n}$ be a closed strongly convex set. Then

$\left(\frac{\Phi_{n-2}(\Omega)}{(n-1)\sigma_{n}}\right)^{2}\geq\left(\frac{\Phi_{n-1}(\Omega)}{\sigma_{n}}\right)^{\frac{2(n-2)}{n-1}}-\left(\frac{\Phi_{n-1}(\Omega)}{\sigma_{n}}\right)^{2}.$

###### Proof.

If $\partial\Omega$ is of class $C^{2}$, then the result follows from Theorem
2.11.

For the general case, let $\Omega$ be a closed strongly convex set. By
Subsection 2.2, we can find closed strongly convex sets $\Omega_{k}$ with
smooth boundaries such that

$\lim_{k}d^{H}(\Omega_{k},\Omega)=0.$

In particular, we have

$\left(\frac{\Phi_{n-2}(\Omega_{k})}{(n-1)\sigma_{n}}\right)^{2}\geq\left(\frac{\Phi_{n-1}(\Omega_{k})}{\sigma_{n}}\right)^{\frac{2(n-2)}{n-1}}-\left(\frac{\Phi_{n-1}(\Omega_{k})}{\sigma_{n}}\right)^{2}.$
(2.6)

Since $\Omega_{k}$ are strongly convex, we have by Subsection 2.3

$\mathcal{R}(\Omega_{k})\geq\frac{\pi}{2}.$

Therefore, we can apply Theorem 2.10, and passing to the limit in (2.6), the
assertion follows. ∎

###  2.4 Isoperimetric inequality

######  Definition  (Minkowski Perimiter).

Let $\Omega\subset\mathbb{S}^{n}$ be a Borel set. We define the _lower
Minkowski content_ as

$\operatorname{Mink}_{-}(\Omega):=\liminf_{s\to
0^{+}}\dfrac{\lvert\Omega^{s}\rvert-\lvert\Omega\rvert}{s}.$

We now state the isoperimetric inequality on spheres in terms of Minkowski
content. The original proof of this result is due to Schmidt in [32] (see also
[30, Theorem 3.15, Theorem 1.52, Theorem 5.18]).

######  Theorem 2.12.

Let $\Omega\subset\mathbb{S}^{n}$ be a measurable set, and let
$B\subset\mathbb{S}^{n}$ be a geodesic ball having the same measure as
$\Omega$. Then,

$\operatorname{Mink}_{-}(\partial
B)\leq\operatorname{Mink}_{-}(\partial\Omega),$

and the equality holds if and only if $\Omega$ is a geodesic ball.

In particular, Subsection 2.3 ensures the following corollary for strongly
convex sets of the sphere.

######  Corollary .

Let $\Omega\subset\mathbb{S}^{n}$ be an open set such that $\bar{\Omega}$ is
strongly convex. Let $B\subset\mathbb{S}^{n}$ be a geodesic ball having the
same measure as $\Omega$, then

$\mathcal{H}^{n-1}(\partial B)\leq\mathcal{H}^{n-1}(\partial\Omega),$

and the equality holds if and only if $\Omega$ is a geodesic ball.

##  3 Proof of the main theorem

In this section, for strongly convex sets $\Omega\subset\mathbb{S}^{n}$, we
denote by $P(\Omega)=\mathcal{H}^{n-1}(\partial\Omega)$.

Let $\Omega\subset\mathbb{S}^{n}$ be an open set with strongly Lipschitz
boundary, then the variational characterization (1.2) is well posed and the
minimizers are weak solutions of (1.1).

######  Remark .

Let $D$ be a geodesic ball of center $q$ and radius $R>0$ in the sphere
$\mathbb{S}^{n}$. We recall that the eigenfunctions relative to the first
eigenvalue $\lambda_{\beta}(D)$ are all proportional. Therefore, by the
rotational symmetry of $D$ and the rotational invariance of the equation
(1.1), we have that all the first eigenfunctions on $D$ are radial. Precisely,
$u(p)=\psi(d(p,q))$ for some function $\psi$ solution to the one-dimensional
problem

$\begin{cases}\psi^{\prime\prime}+(n-1)\cot(r)\psi^{\prime}+\lambda_{\beta}(D)\psi=0&r\in(0,R),\\\\[3.0pt]
\psi^{\prime}(0)=0,\\\\[3.0pt] \psi^{\prime}(R)+\beta\psi(R)=0.\end{cases}$

Moreover letting $\phi(\rho)=\psi(R-\rho)$ we can write $u$ as a function of
the distance from the boundary of the ball $D$, indeed for every $p\in D$

$d(p,\partial D)=R-d(p,q),$

so that $u(p)=\phi(d(p,\partial D))$.

For every $\Omega\subset\mathbb{S}^{n}$, we denote by $R_{\Omega}$ its
inradius, that is

$R_{\Omega}=\max_{p\in\Omega}d(p,\partial\Omega).$

We have the following

######  Lemma .

Let $\Omega\subset\mathbb{S}^{n}$ be a closed, strongly convex set, and let
$\Omega_{t}=(\Omega)_{t}$. Then, for almost every $t\in(0,R_{\Omega})$ the
function $P(\Omega_{t})$ is differentiable and

$-\dfrac{d}{dt}P(\Omega_{t})\geq(n-1)\left(\sigma_{n}^{\frac{2}{n-1}}P(\Omega_{t})^{\frac{2(n-2)}{n-1}}-P(\Omega_{t})^{2}\right)^{\frac{1}{2}}.$
(3.1)

###### Proof.

From the strong convexity of $\Omega$, by Subsection 2.2 we have that the for
every $R_{\Omega}>s>t>0$ the inner parallel sets $\Omega_{t}$ and $\Omega_{s}$
are strongly convex, so that Subsection 2.2 ensures that

$P(\Omega_{s})\leq P(\Omega_{t}).$

In particular, the function $t\mapsto P(\Omega_{t})$ is monotonic decreasing
and hence it is differentiable almost everywhere. Fix $t\in(0,R_{\Omega})$,
for every $s>0$ sufficiently small, by Subsection 2.2, we have that the sets
$(\Omega_{t})^{s}$ are strongly convex. Moreover, since by definition

$(\Omega_{t})^{s}\subseteq\Omega_{t-s},$

and both are strongly convex, we can apply Subsection 2.2 again, so that

$P((\Omega_{t})^{s})\leq P(\Omega_{t-s}).$

In particular, we get for almost every $t\in(0,R_{\Omega})$

$-\dfrac{d}{dt}P(\Omega_{t})=\lim_{s\to
0^{+}}\dfrac{P(\Omega_{t-s})-P(\Omega_{t})}{s}\geq\lim_{s\to
0^{+}}\dfrac{P((\Omega_{t})^{s})-P(\Omega_{t})}{s}={\left.\kern-1.2pt\dfrac{d}{ds}P((\Omega_{t})^{s})\vphantom{\big{|}}\right|_{s=0}}.$

By the Steiner formula (Theorem 2.9), we have

${\left.\kern-1.2pt\dfrac{d}{ds}P((\Omega_{t})^{s})\vphantom{\big{|}}\right|_{s=0}}=\Phi_{n-2}(\Omega_{t}).$

Hence, (3.1) follows from the Alexandrov-Fenchel inequality Subsection 2.3. ∎

In order to prove Theorem 1.1 we need a comparison result that relates
$P((\Omega)_{t})$ and $P((B)_{t})$.

######  Lemma .

Let $f[a,b]\to\mathbb{R}$ be a monotone decreasing function, and let
$g:[a,b]\to\mathbb{R}$ be an absolutely continuous function. Assume that there
exists a Lipschitz function $F:\mathbb{R}\to\mathbb{R}$ such that

$\begin{dcases}f(a)\leq g(a),\\\ f^{\prime}(t)\leq F(f(t))&\text{for a.e.
}t\in(a,b),\\\ g^{\prime}(t)=F(g(t))&\text{for a.e. }t\in(a,b),\end{dcases}$

then $f(t)\leq g(t)$ for every $t\in[a,b]$.

###### Proof.

First we recall that since $f$ is decreasing then for every $t,s\in(a,b]$ such
that $t<s$, (see for instance [4, Corollary 3.29])

$f(s)-f(t)\leq
f(s^{-})-f(t^{+})\leq\int_{t}^{s}f^{\prime}(\rho)\,d\rho\leq\int_{t}^{s}F(f(\rho))\,d\rho,$
(3.2)

where we used the notation $f(s^{\pm})=\lim_{\varepsilon\to
0^{\pm}}f(s+\varepsilon)$. On the other hand, for $g$ we have the equality

$g(s)-g(t)=\int_{t}^{s}g^{\prime}(\rho)\,d\rho=\int_{t}^{s}F(g(\rho))\,d\rho.$
(3.3)

Subtracting (3.3) to (3.2), and letting $w(t)=f(t)-g(t)$, then

$\begin{split}w(s)-w(t)&\leq\int_{t}^{s}\left(F(f(\rho))-F(g(\rho))\right)\,dt\\\\[7.0pt]
&\leq L\int_{t}^{s}\lvert w(\rho)\rvert\,d\rho,\end{split}$ (3.4)

where $L$ is the Lipschitz constant of $F$. We also notice that by the
monotonicity of $f$ and the continuity of $g$ we have

$w(s)\leq w(s^{-})\qquad\qquad\forall s\in(a,b],$ (3.5) $w(s^{+})\leq
w(s)\qquad\qquad\forall s\in[a,b).$ (3.6)

By contradiction, let us assume that for some $t_{0}\in(a,b]$ we have
$w(t_{0})>0$. Then (3.5) ensures that for a suitable $\delta>0$ and for every
$s\in(t_{0}-\delta,t_{0}]$ we have $w(s)>0$. Let

$\tau=\sup\set{t\in[a,t_{0})}{w(t)\leq 0},$

so that $a\leq\tau\leq t_{0}-\delta$, and

$w(s)>0\qquad\forall s\in(\tau,t_{0}].$ (3.7)

By definition of $\tau$, we have

$w(\tau^{+})\geq 0.$ (3.8)

We claim that $w(\tau)=0$. Indeed, if $\tau=a$, then the initial condition
yields

$w(a)\leq 0,$

and by (3.8), joint with (3.6), we get $0\leq w(a^{+})\leq w(a)\leq 0$. If
$\tau>a$, then, by definition of $\tau$,

$w(\tau^{-})\leq 0.$

Using (3.8) joint with (3.5) and (3.6), we also have $0\leq w(\tau^{+})\leq
w(\tau^{-})\leq 0$, and the claim is proved.

Therefore, since $w(\tau)=0$, (3.4) reads as follows: for every
$s\in(\tau,t_{0})$ we have

$w(s)\leq L\int_{\tau}^{s}w(\rho)\,d\rho.$

By the integral form of the Gronwall inequality (see for instance [22, Lemma
3.2], which is a particular case of [23, Theorem 3.1]) , we get $w\leq 0$ in
$[\tau,t_{0}]$, which is in contradiction with (3.7). ∎

We are now able to prove the main theorem.

###### Proof of Theorem 1.1.

Let $D$ be a strongly convex geodesic ball such that

$P(\Omega)=P(D),$

and let $R$ be its radius. The isoperimetric inequality (Subsection 2.4) and
the fact that both $D$ and $\Omega$ are contained in a hemisphere, ensure that
$\lvert\Omega\rvert\leq\lvert D\rvert$. Since $R_{\Omega}$ is the radius of
the biggest ball contained in $\Omega$, we also obtain $R_{\Omega}\leq R$, and
the equality holds if and only if $\Omega$ is a ball of radius $R_{\Omega}$.

For every $t\in(0,R_{\Omega})$, let

$\Omega_{t}=(\Omega)_{t},\qquad\text{and}\qquad D_{t}=(D)_{t}$

be the inner parallel sets of $\Omega$ and $D$ respectively. Then from Section
3 we have that

$\dfrac{d}{dt}P(\Omega_{t})\leq-(n-1)\left(\sigma_{n}^{\frac{2}{n-1}}P(\Omega_{t})^{\frac{2(n-2)}{n-1}}-P(\Omega_{t})^{2}\right)^{\frac{1}{2}},$

while, by direct computation, the same estimate holds for the perimeter of
$D_{t}$ with the equality sign

$\dfrac{d}{dt}P(D_{t})=-(n-1)\left(\sigma_{n}^{\frac{2}{n-1}}P(D_{t})^{\frac{2(n-2)}{n-1}}-P(D_{t})^{2}\right)^{\frac{1}{2}}.$

The comparison lemma (Section 3) ensures that for every $t\in(0,R_{\Omega})$

$P(\Omega_{t})\leq P(D_{t}).$ (3.9)

Let $u$ be an eigenfunction on $D$ and let $\phi\colon[0,R]\to\mathbb{R}$ be
as in Section 3, then for every $p\in D$

$u(p)=\phi(d(p,\partial D)).$

For every $p\in\Omega$, let us define

$v(p)=\phi(d(p,\partial\Omega)),$

so that $v\in H^{1}(\Omega)$, and

$\lambda_{\beta}(\Omega)\leq\dfrac{\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,d\mu+\beta\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\Omega}v^{2}\,d\mu}.$

By direct computation, we have that

$\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}=\phi^{2}(0)P(\Omega)=\int_{\partial
D}u^{2}\,d\mathcal{H}^{n-1}.$

While, using coarea formula (Theorem 2.1) with $f(p)=d(p,\partial\Omega)$ and
(3.9), we have

$\int_{\Omega}v^{2}\,d\mu=\int_{0}^{R_{\Omega}}\phi^{2}(t)P(\Omega_{t})\,dt\leq\int_{0}^{R_{\Omega}}\phi^{2}(t)P(D_{t})\,dt\leq\int_{D}u^{2}\,d\mu,$
(3.10)

and

$\int_{\Omega}\lvert\nabla
v\rvert^{2}\,d\mu=\int_{0}^{R_{\Omega}}(\phi^{\prime}(t))^{2}P(\Omega_{t})\,dt\leq\int_{0}^{R_{\Omega}}(\phi^{\prime}(t))^{2}P(D_{t})\,dt\leq\int_{D}\lvert\nabla
u\rvert^{2}\,d\mu.$

Then

$\int_{\Omega}\lvert\nabla
v\rvert^{2}\,d\mu+\beta\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}\leq\int_{D}\lvert\nabla
u\rvert^{2}+\beta\int_{\partial D}u^{2}\,d\mathcal{H}^{n-1}<0.$

Hence,

$\lambda_{\beta}(\Omega)\leq\dfrac{\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,d\mu+\beta\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\Omega}v^{2}\,d\mu}\leq\dfrac{\displaystyle\int_{D}\lvert\nabla
u\rvert^{2}\,d\mu+\beta\int_{\partial
D}u^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{D}u^{2}\,d\mu}=\lambda_{\beta}(D).$

Finally, if the equality $\lambda_{\beta}(\Omega)=\lambda_{\beta}(D)$ holds,
then the equality in (3.10) gives that $R_{\Omega}=R$, which implies that
$\Omega$ is a geodesic ball of radius $R_{\Omega}$. ∎

Following the approach of [3] we now prove Theorem 1.2.

###### Proof of Theorem 1.2.

As in the proof of Theorem 1.1, let $\phi\colon[0,R]\to\mathbb{R}$ be such
that $u(p)=\phi(d(p,\partial D))$ and let $v(p)=\phi(d(p,\partial\Omega))$. In
order to obtain (1.6) we can better estimate the $L^{2}$-norm of the test $v$.
Indeed,

$\begin{split}\int_{\Omega}v^{2}\,d\mu=&\int_{0}^{R_{\Omega}}\phi^{2}(t)P(\Omega_{t})\,dt\leq\int_{0}^{R_{\Omega}}\phi^{2}(t)P(D_{t})\,dt\\\\[15.0pt]
=&\int_{0}^{R}\phi^{2}(t)P(D_{t})\,dt-\int_{R_{\Omega}}^{R}\phi^{2}(t)P(D_{t})\,dt\leq\int_{D}u^{2}\,d\mu-
u_{m}^{2}\int_{R_{\Omega}}^{R}P(D_{t})\,dt\\\\[15.0pt]
\leq&\int_{\Omega}u^{2}\,d\mu-u_{m}^{2}(\lvert
D\rvert-\lvert\Omega\rvert)=\int_{\Omega}u^{2}\,d\mu\left(1-\dfrac{u_{m}^{2}}{\lVert
u\rVert_{L^{2}(D)}^{2}}(\lvert D\rvert-\lvert\Omega\rvert)\right),\end{split}$

where we have used that, by definition of inradius, for a suitable ball
$B_{R_{\Omega}}$ of radius $R_{\Omega}$ we have
$B_{R_{\Omega}}\subseteq\Omega$. Therefore, computations analogous to the ones
done in Theorem 1.1 lead to

$\begin{split}\lambda_{\beta}(\Omega)&\leq\dfrac{\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,d\mu+\beta\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\Omega}v^{2}\,d\mu}\\\\[15.0pt]
&\leq\dfrac{\displaystyle\int_{D}\lvert\nabla
u\rvert^{2}\,d\mu+\beta\int_{\partial
D}u^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{D}u^{2}\,d\mu\left(1-\dfrac{u_{m}^{2}}{\lVert
u\rVert_{L^{2}(D)}^{2}}(\lvert D\rvert-\lvert\Omega\rvert)\right)}\\\\[15.0pt]
&=\lambda_{\beta}(D)\left(1-\dfrac{u_{m}^{2}}{\lVert
u\rVert_{L^{2}(D)}^{2}}(\lvert
D\rvert-\lvert\Omega\rvert)\right)^{-1}.\end{split}$

So that, reordering the terms, (1.6) is proved. ∎

##  4 Further remarks

In this section, we show that the same arguments used to prove Theorem 1.1
cannot be used for strongly convex sets in the hyperbolic setting. Even though
it is possible to generalize the Alexandrov-Fenchel inequalities to the
hyperbolic space under strict convexity assumptions (see for instance [36,
Theorem 1.1]), the main difficulty here is to extend Subsection 2.2 to the
hyperbolic space $\mathbb{H}^{n}$. In particular, we can construct convex sets
for which the inner parallel sets are not convex. To show an example, let us
fix some notation. Let

$\lVert x\rVert_{e}=\sqrt{\langle x,x\rangle}$

be the Euclidean norm, and let $\mathbb{H}^{n}$ be represented in the Poincaré
half-space model:

$\displaystyle\mathbb{H}^{n}=\Set{(\hat{x},x_{n})\in\mathbb{R}^{n}}{x_{n}>0},$
$\displaystyle g_{x}(v,w)=\frac{\langle v,w\rangle}{x_{n}^{2}},$
$\displaystyle d(x,y)=2\operatorname{arcsinh}\left(\frac{\lVert
x-y\rVert_{e}}{2\sqrt{x_{n}y_{n}}}\right).$

We also recall the shape of the geodesics in $\mathbb{H}^{n}$: let
$p,q\in\mathbb{H}^{n}$, if $\hat{p}=\hat{q}$, then the geodesic $\gamma_{pq}$
connecting the two is the vertical line passing through $p$ and $q$; if
$\hat{p}\neq\hat{q}$, then the geodesic $\gamma_{pq}$ connecting $p$ and $q$
is the unique circular arc touching orthogonally the plane $\\{x_{n}=0\\}$. We
could describe the circular arc (non-parametrized by arc length) as

$\displaystyle\gamma_{pq}:[t_{p},t_{q}]$
$\displaystyle\longrightarrow\mathbb{H}^{n}$ (4.1) $\displaystyle t$
$\displaystyle\longmapsto x_{0}+R\left(t\hat{w},\sqrt{1-t^{2}}\right),$

where $x_{0}=(\hat{x}_{0},0)$ is the center of the circular arc, $R=\lVert
x_{0}-p\rVert_{e}$ is the radius of the arc,

$\hat{w}=\frac{\hat{p}-\hat{q}}{\lVert\hat{p}-\hat{q}\rVert_{e}},$

and

$\displaystyle[t_{p},t_{q}]\subset(-1,1),$
$\displaystyle\gamma_{pq}(t_{p})=p\qquad\qquad\gamma_{pq}(t_{q})=q.$

We now divide the construction into three simple steps.

Step 1: Consider the cylinder

$C=\Set{(\hat{x},x_{n})\in\mathbb{R}^{n}}{\begin{gathered}\lVert\hat{x}\rVert_{e}\leq
1\end{gathered}}.$

C is convex:

consider $p,q\in C$, and let $\gamma_{pq}$ be the geodesic connecting the
points. If $\hat{p}=\hat{q}$, then we can represent the geodesic (non-
parametrized by arc length) as

$\gamma_{pq}(t)=(\hat{p},t),$

and obviously $\gamma_{pq}(t)\in C$ for every $t$. If $\hat{p}\neq\hat{q}$,
then for the geodesic $\gamma_{pq}=(\hat{\gamma}_{pq},\gamma_{pq}^{n})$ we
have that $\hat{\gamma}_{pq}([t_{p},t_{q}])$ is the segment joining $\hat{p}$
and $\hat{q}$ in $\mathbb{R}^{n-1}$, so that

$\lVert\hat{\gamma}_{pq}(t)\rVert_{e}\leq\max\\{\lVert\hat{p}\rVert_{e},\lVert\hat{q}\rVert_{e}\\}\leq
1,$

for every $t\in[t_{p},t_{q}]$.

Step 2. For any fixed vertical line
$r(\hat{x}_{0})=\set{(\hat{x},x_{n})}{\hat{x}=\hat{x}_{0}}$, the level sets of
the distance from $r(\hat{x}_{0})$ are cones:

it is sufficient to notice that for every point $(\hat{x}_{0},x_{n})$ the
geodesics orthogonal to $r(\hat{x}_{0})$ in $(\hat{x}_{0},x_{n})$ are all
contained in the hemisphere of radius $x_{n}$ centered in
$x_{0}=(\hat{x}_{0},0)$, so that, with a direct computation,

$(r(\hat{x}_{0}))^{t}=\Set{x\in\mathbb{R}^{n}}{d(x,r_{0})\leq
t}=\Set{(\hat{x},x_{n})\in\mathbb{R}^{n}}{\lVert\hat{x}-\hat{x}_{0}\rVert_{e}\leq\sinh(t)\,x_{n}}.$

Step 3: the inner parallel sets $(C)_{\delta}$ are not convex for every choice
of $\delta>0$.

Indeed, notice that

$(C)_{\delta}=C\cap\bigcap_{\lVert\hat{x}_{0}\rVert_{e}=1}\overline{\mathbb{H}^{n}\setminus(r(\hat{x}_{0}))^{\delta}}=\Set{(\hat{x},x_{n})\in\mathbb{R}^{n}}{\begin{gathered}x_{n}>0,\\\
\lVert\hat{x}\rVert_{e}\leq 1-\sinh(\delta)\,x_{n}.\end{gathered}}$

If for instance we take $q\in(C)_{\delta}$ such that
$\lVert\hat{q}\rVert_{e}=1-\sinh(\delta)q_{n}$ and $p=(0,1/\sinh(\delta)))$,
then the minimal geodesic $\gamma_{pq}=(\hat{\gamma},\gamma^{n})$ connecting
$p$ and $q$ lies outside the cone $(C)_{\delta}$: we can write, for
$t\in[t_{p},t_{q}]$,

$\gamma(t)=x_{0}+R\left(t\hat{w},\sqrt{1-t^{2}}\right)$

as defined in (4.1); notice that by concavity

$\begin{split}\gamma^{n}(t)&>\gamma^{n}(t_{p})+\frac{t-t_{p}}{t_{q}-t_{p}}(\gamma^{n}(t_{q})-\gamma^{n}(t_{p}))\\\\[7.0pt]
&=p_{n}+\frac{t-t_{p}}{t_{q}-t_{p}}(q_{n}-p_{n})\qquad\qquad\forall
t\in(t_{p},t_{q}),\end{split}$ (4.2)

and that, since $\hat{p}=0$, then $\hat{w}$ and $\hat{x}_{0}$ are
proportional, then $\lVert\hat{\gamma}(t)\rVert_{e}$ is linear in $t$, so that
it is of the form

$\lVert\hat{\gamma}_{pq}(t)\rVert_{e}=\lVert\hat{p}\rVert_{e}+\frac{t-t_{q}}{t_{p}-t_{q}}\lVert\hat{q}\rVert_{e}.$
(4.3)

Using the fact that $p,q\in\partial(C)_{\delta}$, and in particular (4.2) and
(4.3), which implies that the geodesic is not contained in $(C)_{\delta}$.

#### Acknowledgements

The five authors are members of Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta
Matematica (INdAM). The authors Paolo Acampora, Emanuele Cristoforoni and
Carlo Nitsch were partially supported by the project GNAMPA 2023: "Symmetry
and asymmetry in PDEs", CUP_E53C22001930001. The authors Paolo Acampora and
Emanuele Cristoforoni were partially supported by the 2024 project GNAMPA
2024: "Modelli PDE-ODE nonlineari e proprieta’ di PDE su domini standard e
non-standard", CUP_E53C23001670001. The author Antonio Celentano was partially
supported by the INdAM-GNAMPA project 2023 "Modelli matematici di EDP per
fluidi e strutture e proprieta’ geometriche delle soluzioni di EDP", cod.
CUP_E53C22001930001, and by the INdAM-GNAMPA project 2024 "Problemi
frazionari: proprietà quantitative ottimali, simmetria, regolarità", cod.
CUP_E53C23001670001.

We would like to thank Marco Pozzetta for his suggestions and fruitful
discussions about the topic.

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Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
studi di Napoli Federico II, Via Cintia, Complesso Universitario Monte S.
Angelo, 80126 Napoli, Italy.

E-mail address, P. Acampora<EMAIL_ADDRESS>
E-mail address, A. Celentano<EMAIL_ADDRESS>
E-mail address, C. Nitsch<EMAIL_ADDRESS>
E-mail address, C. Trombetti<EMAIL_ADDRESS>
Mathematical and Physical Sciences for Advanced Materials and Technologies,
Scuola Superiore Meridionale, Largo San Marcellino 10, 80126, Napoli, Italy.

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