# Fractional perimeters on the sphere

Andreas Kreuml, Olaf Mordhorst

###### Abstract

This note treats several problems for the fractional perimeter or
$s$-perimeter on the sphere. The spherical fractional isoperimetric inequality
is established. It turns out that the equality cases are exactly the spherical
caps. Furthermore, the convergence of fractional perimeters to the surface
area as $s\nearrow 1$ is proven. It is shown that their limit as
$s\searrow-\infty$ can be expressed in terms of the volume.

##  1 Introduction

In the Euclidean setting the fractional $s$-perimeter, $0<s<1$, of a Borel set
$E\subseteq\mathbb{R}^{n}$ is defined by the double integral

$P_{s}(E)=\int_{E}\int_{\mathbb{R}^{n}\backslash
E}{\frac{1}{|x-y|^{n+s}}}\,\mathrm{d}y\,\mathrm{d}x,$

where $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^{n}$. Fractional
perimeters can be seen as a special case of fractional Sobolev norms and their
first systematic study was initiated by Caffarelli, Roquejoffre & Savin in
[7]. It is natural to extend this notion to the setting of Riemannian
manifolds. Let $(M,g)$ be a compact, connected Riemannian manifold of
dimension $n$, $d$ the geodesic distance and $\text{d}V_{g}$ the volume
element induced by the Riemannian metric $g$. In [14] the fractional
$s$-perimeter of a Borel set $E\subseteq M$ is defined as

$P_{s}(E):=\int_{E}\int_{M\backslash
E}\frac{1}{d(x,y)^{n+s}}\,\mathrm{d}V_{g}(y)\,\mathrm{d}V_{g}(x).$

Since there will be no confusion, we use the same notation $P_{s}$ for both
the Euclidean and Riemannian setting.

In the Euclidean setting, fractional perimeters are interesting objects in
geometric measure theory since they interpolate between the volume and the
surface area. Conceptually, fractional perimeters are a coarsened version of
the perimeter functional $P$ due to de Giorgi [9]. Another fundamental
property is that they provide a family of isoperimetric-type inequalities.

The purpose of this paper is to transfer these geometric principles to the
study of fractional perimeters on the sphere. We give the precise statements
later on. We like to point out that a good portion of the facts presented here
rely on the special structure of the sphere. In contrast to the Euclidean
space, the sphere lacks a vector space structure but on the other hand has the
property that it ”looks everywhere the same”, for example in the sense of
curvature or a symmetry group acting transitively on it. However, the
interpolation between volume and surface area by fractional perimeters does
not work completely analogously to the Euclidean case and we think that a
transfer to more general manifolds is not possible. Yet, other examples of
manifolds with good symmetry properties and similar geometric features would
be of greater interest.

We like to mention that a quite similar theory of geometric nature exists for
the $L_{p}$-affine surface area on convex bodies. The $L_{p}$-affine surface
area was introduced in [17] and interpolates between the classical affine
surface area due to Blaschke and the volume of the polar of a convex body (see
[24] for an overview). They also fulfill isoperimetric-type inequalities
([18], [19], [25]). Recently, the notion of affine surface area was extended
to the case of convex bodies on the sphere ([3]).

For the scope of this paper we use the following more suitable notation for
fractional perimeters on the sphere. The $n$-dimensional sphere is denoted by
$\mathbb{S}^{n}\subseteq\mathbb{R}^{n+1}$, for $E\subseteq\mathbb{S}^{n}$ we
put $E^{c}:=\mathbb{S}^{n}\backslash E$ and the $k$-dimensional Hausdorff
measure is denoted by $\mathcal{H}^{k}$. The fractional $s$-perimeter for
Borel sets $E$ is then given by

$P_{s}(E)=\int_{E}\int_{E^{c}}\frac{1}{d(x,y)^{n+s}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x).$
(1.1)

Since the sphere is compact it is also possible to study $s$-perimeters for
negative values of $s$, i.e. we consider $s\in(-\infty,1)$ in this paper. Note
that for $s\leq-n$ there are no singularities in the integrand and it can
easily be seen that the integrals also converge whenever $-n<s<0$.

The first part deals with the isoperimetric inequality for spherical
fractional perimeters. For every Euclidean ball $B\subseteq\mathbb{R}^{n}$ and
every Borel set $E\subseteq\mathbb{R}^{n}$ with
$\mathcal{H}^{n}(E)=\mathcal{H}^{n}(B)$ we have the classical isoperimetric
inequality in $\mathbb{R}^{n}$,

$P(E)\geq P(B),$ (1.2)

and for every $s\in(0,1)$ the fractional isoperimetric inequality in
$\mathbb{R}^{n}$,

$P_{s}(E)\geq P_{s}(B),$ (1.3)

where in both inequalities equality holds only for balls of the same
$n$-dimensional Hausdorff measure up to nullsets. General references for the
discussion and proof of (1.2) and (1.3) are [20] and [10] respectively. There
are nowadays a lot of isoperimetric inequalities in different settings, for
example for the Gauss measure (see [4], [23]), on the sphere (first discovered
by Paul Lévy) and for anisotropic fractional perimeters (see [13], [15]). We
apply a general rearrangement inequality established by Beckner [2] to
spherical fractional perimeters and derive that for every spherical cap
$C\subseteq\mathbb{S}^{n}$ and every Borel set $E$ with
$\mathcal{H}^{n}(E)=\mathcal{H}^{n}(C)$ and every $s\in(-n,1)$ we have

$P_{s}(E)\geq P_{s}(C)$ (1.4)

with equality only for spherical caps of same $n$-dimensional Hausdorff
measure up to nullsets. For $-\infty<s<-n$ we show a reverse isoperimetric-
type inequality.

The second part deals with the convergence as $s$ tends to $1$ from below. In
the Euclidean setting a result by Dávila [8] based on the work of Bourgain,
Brézis & Mironescu (see [5], and also [6]) states that if
$E\subseteq\mathbb{R}^{n}$ is a Borel set, then

$\displaystyle\lim_{s\nearrow
1}(1-s)P_{s}(E)=\mathcal{H}^{n-1}(\mathbb{B}^{n-1})P(E)$ (1.5)

where $\mathbb{B}^{n-1}$ is the closed $(n-1)$-dimensional Euclidean unit
ball. If $E$ has sufficiently regular boundary, $P$ just coincides with the
usual surface area.

An analogous result was already shown for more general Riemannian manifolds
than the sphere in [14]. Hence, this result actually does not depend on the
particular structure of the sphere. The proof technique is based on cutting
the manifold into small pieces, approximating every piece by a Euclidean space
and using ideas from [5] and [8]. Yet, we present a different proof based on
ideas developed by Ludwig in [15] and [16] for anisotropic fractional
perimeters and norms. Ludwig uses tools from integral geometry, namely the
Blaschke-Petkantschin formula. Here, we use a spherical version of the
Blaschke-Petkantschin formula which follows from a kinematic formula by
Arbeiter & Zähle [1] and for which a more accesible proof was given recently
by Hug & Thäle [12]. Thus, using properties of the sphere we present a more
organized and concise approach. However, we show the convergence result on the
sphere only for polyconvex sets. In Ludwig’s proof a result of Wieacker [26,
Theorem 1] is used which is only available in the Euclidean setting. Since the
general result is already proven in [14], we decided not to include a proof of
a spherical version of Wieacker’s result since our objective is of conceptual
nature.

As a small remark we like to point out that the classical isoperimetric
inequality on the sphere follows from the fractional variant (1.4) and the
aforementioned convergence result.

The main result of this paper deals with the convergence of spherical
fractional perimeters and seminorms as $s\searrow-\infty$. We show that if
$f\in L_{p}(\mathbb{S}^{n})$ where $1\leq p<\infty$ and
$\tilde{d}(x,y):=\frac{d(x,y)}{\pi}$ is the normalized geodesic distance
between $x,y\in\mathbb{S}^{n}$, then

$\lim_{s\searrow-\infty}(-s)^{n}\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}\frac{|f(x)-f(y)|^{p}}{\tilde{d}(x,y)^{n+sp}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)=c_{n,p}\int_{\mathbb{S}^{n}}|f(x)-f(-x)|^{p}\,\mathrm{d}\mathcal{H}^{n}(x),$

where $c_{n,p}$ is a constant depending on $n$ and $p$ whose explicit value is
given in Theorem 5. In particular, the normalized fractional perimeter, where
the distance $d$ is replaced by its normalized version $\tilde{d}$, has a zero
of order $n$ at $s=-\infty$, and

$\lim_{s\searrow-\infty}(-s)^{n}\int_{E}\int_{E^{c}}\frac{1}{\tilde{d}(x,y)^{n+s}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)=c_{n,1}\mathcal{H}^{n}((-E)\,\cap\,(E^{c}))$

whenever $E\subseteq\mathbb{S}^{n}$ is a Borel set. This result is in a
similar spirit as a result by Maz’ya & Shaposhnikova [21] who showed that if
$f$ lies in the fractional Sobolev space $W^{s_{0},p}(\mathbb{R}^{n})$ for a
certain $s_{0}\in(0,1)$, then

$\lim_{s\searrow
0}s\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+sp}}\,\mathrm{d}y\,\mathrm{d}x=d_{n,p}\int_{\mathbb{R}^{n}}|f(x)|^{p}\,\mathrm{d}x$
(1.6)

where the constant $d_{n,p}$ depends on $n$ and $p$. We like to emphasize that
the corresponding limit procedure of (1.6) on the sphere does not lead to the
same result since it is $0$ for smooth functions (see Section 5).

The paper is organized as follows. In the second section we provide some
notions and definitions. In the third section we show the aforementioned
isoperimetric inequality. In the fourth section we provide some lemmata of
integral geometric nature and prove the convergence of spherical fractional
perimeters to the surface area as $s\nearrow 1$. In the fifth section we
consider the limit of spherical fractional perimeters as $s\searrow-\infty$.

##  2 Definitions and Notation

For two vectors $x=(x_{1},\dots,x_{n+1})$ and $y=(y_{1},\dots,y_{n+1})$ in
$\mathbb{R}^{n+1}$ the Euclidean inner product is given by $x\cdot
y=\sum_{j=1}^{n+1}x_{j}y_{j}$. It induces the Euclidean norm $|x|=\sqrt{x\cdot
x}$. The Euclidean unit sphere $\mathbb{S}^{n}\subseteq\mathbb{R}^{n+1}$ is
then given by $\mathbb{S}^{n}=\left\\{x\in\mathbb{R}^{n+1}:|x|=1\right\\}$. It
inherits the topology from $\mathbb{R}^{n+1}$ and we write $\partial E$ for
the boundary of the set $E\subseteq\mathbb{S}^{n}$ and
$E^{c}=\mathbb{S}^{n}\backslash E$ for the complement of $E$ in the sphere.
For $x\in\mathbb{R}^{n+1}$ and $r>0$ we denote by
$B_{r}^{n+1}(x):=\left\\{y\in\mathbb{R}^{n+1}:|x-y|<r\right\\}$ the open
Euclidean ball around $x$ with radius $r$ and simply write
$B_{r}^{n+1}:=B_{r}^{n+1}(0)$ for balls centered at the origin. The indicator
function of a set $E$ is the function defined by $\mathds{1}_{E}(x)=1$ for
$x\in E$ and $\mathds{1}_{E}(x)=0$ for $x\notin E$. Furthermore we denote the
$k$-dimensional Hausdorff measure in $\mathbb{R}^{n+1}$ by $\mathcal{H}^{k}$,
$k\in\left\\{0,\dots,n+1\right\\}$, such that the restriction of
$\mathcal{H}^{n}$ to $\mathbb{S}^{n}$ is equal to the spherical Lebesgue
measure on the Borel sets of $\mathbb{S}^{n}$. We write $\omega_{n+1}$ for the
surface area of the $n$-dimensional unit sphere, i.e.
$\omega_{n+1}=\mathcal{H}^{n}(\mathbb{S}^{n})$.

We denote the Grassmannian of 2-dimensional subspaces in $\mathbb{R}^{n+1}$ by
$G(n+1,2)$ and equip it with the Haar measure $\mathrm{d}L$ such that
$\int_{G(n+1,2)}\,\mathrm{d}L=1$.

The geodesic distance $d(x,y)$ between two points $x,y\in\mathbb{S}^{n}$ is
defined as

$d(x,y):=\inf\ell(\gamma)\quad(\in[0,\pi]),$

where the infimum ranges over all piecewise $C^{1}$-curves connecting $x$ and
$y$ and $\ell(\gamma)$ denotes the length of the curve $\gamma$. The infimum
is attained for great circle arcs, which lie in the intersection of any
2-dimensional plane through the origin containing $x$ and $y$ with
$\mathbb{S}^{n}$. We further put

$d(E,x):=\inf_{y\in E}d(y,x)$

for $E\subseteq\mathbb{S}^{n}$. A non-empty subset $K\subseteq\mathbb{S}^{n}$
is called a (spherically) convex body if the cone

$\text{pos}(K):=\left\\{\lambda x:\lambda\geq 0,x\in K\right\\}$

generated by $K$ is a closed convex subset of $\mathbb{R}^{n+1}$. Note that
for any pair of points $x,y\in K$ the shorter geodesic line segment connecting
$x$ and $y$ lies entirely in $K$. We further remark that for each
$x\in\mathbb{S}^{n}$ such that $0\leq d(K,x)<\pi/2$ there is a unique point
$p(K,x)$ in $K$ that is nearest to $x$. We say that the set
$E\subseteq\mathbb{S}^{n}$ is _polyconvex_ if it can be written as a finite
union of convex bodies.

A convex body $P\subseteq\mathbb{S}^{n}$ is called a (spherical) polytope, if
its cone $\text{pos}(P)$ is the intersection of finitely many halfspaces. We
call $F$ a $k$-face of $P$, $k\in\left\\{0,\dots,n\right\\}$, if
$F=\tilde{F}\,\cap\,\mathbb{S}^{n}$, where $\tilde{F}$ is a $(k+1)$-face of
the polyhedral cone $\text{pos}(P)$.

If $K\subseteq\mathbb{S}^{n}$ is a convex body, its polar body $K^{\circ}$ is
defined by

$K^{\circ}:=\left\\{x\in\mathbb{S}^{n}:x\cdot y\leq 0\text{ for all }y\in
K\right\\}.$

The normal cone $N(K,x)$ of $K$ at $x\in\partial K$ is then defined by

$N(K,x):=\left\\{y\in K^{\circ}:x\cdot y=0\right\\}.$

For our integral-geometric treatment of the fractional perimeter we will use
spherical curvature measures, which satisfy a local spherical Steiner formula.
For a convex body $K\subseteq\mathbb{S}^{n}$, a Borel set
$A\subseteq\mathbb{S}^{n}$ and $0<\varepsilon<\pi/2$ we put

$M_{\varepsilon}(K,A):=\left\\{x\in\mathbb{S}^{n}:d(K,x)\leq\varepsilon,\
p(K,x)\in A\right\\}.$

Then the curvature measures
$\varphi_{0}(K,\cdot),\dots,\varphi_{n-1}(K,\cdot)$ are the uniquely
determined Borel measures on $\mathbb{S}^{n}$ such that for all Borel sets
$A\subseteq\mathbb{S}^{n}$ and $0<\varepsilon<\pi/2$

$\mathcal{H}^{n}(M_{\varepsilon}(K,A))=\sum_{j=0}^{n-1}g_{n,j}(\varepsilon)\varphi_{j}(K,A),$

where

$g_{n,j}(\varepsilon)=\omega_{j+1}\omega_{n-j}\int_{0}^{\varepsilon}\cos^{j}t\,\sin^{n-j-1}t\,\mathrm{d}t,$

see e.g. [22, Theorem 6.5.1] for generalized curvature measures.

##  3 The Spherical Isoperimetric Inequality for $s$-Perimeters

Let $v\in\mathbb{S}^{n}$ and $0\leq r<\infty$. We define the open spherical
cap $C(v,r)$ by

$C(v,r)=\\{w\in\mathbb{S}^{n}:d(v,w)<r\\}.$

Note that $C(v,0)=\emptyset$, $C(v,\pi)=\mathbb{S}^{n}\backslash\\{-v\\}$ and
$C(v,r)=\mathbb{S}^{n}$ if $r>\pi$. We show a spherical isoperimetric
inequality for $s$-perimeters if $s>-n$ and a reverse isoperimetric-type
inequality for $s<-n$.

Theorem 3.1.  Let $E\subseteq\mathbb{S}^{n}$ be a Borel set and $C$ a
spherical cap with $\mathcal{H}^{n}(E)=\mathcal{H}^{n}(C)$. Then

$\displaystyle P_{s}(E)\geq P_{s}(C)$ (3.1)

for $-n<s<1$ and

$\displaystyle P_{s}(E)\leq P_{s}(C)$ (3.2)

for $-\infty<s<-n$. Equality is attained if and only if $E$ is itself a
spherical cap up to a $\mathcal{H}^{n}$-nullset.

Note that $P_{-n}(E)$ does not change for all $E$ of same measure. We present
this fact in the proof of the theorem.

It is easy to see that the theorem can be reformulated as follows: Let
$0<\alpha\leq\omega_{n+1}$. Then there is a constant $\gamma_{n,s,\alpha}$
such that for every Borel set $E\subseteq\mathbb{S}^{n}$ with
$\alpha=\mathcal{H}^{n}(E)$ we have

$P_{s}(E)\geq\gamma_{n,s,\alpha}\mathcal{H}^{n}(E)=\gamma_{n,s,\alpha}\alpha$

if $-n<s<1$ and

$P_{s}(E)\leq\gamma_{n,s,\alpha}\mathcal{H}^{n}(E)=\gamma_{n,s,\alpha}\alpha$

if $-\infty<s<-n$. The constant $\gamma_{n,s,\alpha}$ is given by
$\gamma_{n,s,\alpha}=\frac{P_{s}(C)}{\alpha}$, where
$C\subseteq\mathbb{S}^{n}$ is a spherical cap with
$\mathcal{H}^{n}(C)=\alpha$. Equality is attained if and only if $E$ is a
spherical cap up to a $\mathcal{H}^{n}$-nullset. It is easy to show that

$\lim\limits_{\alpha\nearrow\omega_{n+1}}\gamma_{n,s,\alpha}=0$

for every $-n<s<1$. Hence, one cannot expect to have a uniform constant
$\gamma_{n,s}$ in this case.

In order to prove the theorems we use rearrangement inequalities with respect
to a fixed center of symmetry $e\in\mathbb{S}^{n}$ on the sphere. We use the
same notations as for rearrangements in the Euclidean setting as there will be
no confusions. The function $a:[0,\pi]\rightarrow[0,\omega_{n+1}]$,
$a(r)=\mathcal{H}^{n}(C(v,r))$ does not depend on the choice of
$v\in\mathbb{S}^{n}$ and is strictly increasing and bijective. For a Borel set
$E\subseteq\mathbb{S}^{n}$ the spherical volume radius $r_{\sigma}$ is defined
by

$r_{\sigma}(E)=a^{-1}(\mathcal{H}^{n}(E)).$

and the spherical rearrangement of $E$ by $E^{*}=C(e,r_{\sigma}(E))$. Let
$f:\mathbb{S}^{n}\rightarrow\mathbb{R}$ be a measurable function. The
spherical rearrangement of $f$ is denoted by $f^{*}$ and is defined by the
layer cake representation $f^{*}:\mathbb{S}^{n}\rightarrow\mathbb{R}_{\geq
0}$, $f^{*}(v)=\int_{0}^{\infty}\mathds{1}_{\\{|f|>t\\}^{*}}(v)\mathrm{d}t$.
The following rearrangement inequality on the sphere can be found in [2,
Theorem 3].

Theorem 3.2.  Let $\varphi,k$ and $\rho$ be non-negative functions defined on
$[0,\infty)$ such that

  1. 1.

$\varphi(0)=0$, $\varphi$ is convex and monotonically increasing,
$\varphi^{\prime\prime}\geq 0$ and $t\mapsto t\varphi^{\prime}(t)$ is convex,

  2. 2.

$k$ is monotonically decreasing, and

  3. 3.

$\rho$ is monotonically increasing.

Then for measurable functions $f$ and $g$ on $\mathbb{S}^{n}$

$\displaystyle\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}$
$\displaystyle\varphi\left(\frac{|f(x)-g(y)|}{\rho(d(x,y))}\right)k(d(x,y))\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)$
$\displaystyle\geq$
$\displaystyle\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}\varphi\left(\frac{|f^{*}(x)-g^{*}(y)|}{\rho(d(x,y))}\right)k(d(x,y))\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x).$

If $k$ is strictly decreasing and $\varphi$ is strictly convex, then equality
holds if and only if $f(x)=\lambda f^{*}(\theta x)$ and $g(x)=\lambda
g^{*}(\theta x)$ for a.e. $x\in\mathbb{S}^{n}$, where
$\lambda\in\left\\{+1,-1\right\\}$ and $\theta\in SO(n+1)$.

Proof of Theorem 3. For a Borel set $E\subseteq\mathbb{S}^{n}$ note that

$2\cdot
P_{s}(E)=\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}|\mathds{1}_{E}(x)-\mathds{1}_{E}(y)|^{p}d(x,y)^{-(n+s)}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)$

where $p>1$ is arbitrary.

For $-n<s<1$ apply Theorem 3 with $\varphi(t)=t^{p}$,
$k(t)=\frac{1}{(1+t)^{n+s}}$ and $\rho(t)=(\frac{t}{1+t})^{\frac{n+s}{p}}$.
Note further that

$P_{-n}(E)=\int_{E}\int_{E^{c}}1\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)=\mathcal{H}^{n}(E)(\mathcal{H}^{n}\left(\mathbb{S}^{n})-\mathcal{H}^{n}(E)\right)$

and this quantity is always the same for $E$’s of same Hausdorff measure,
especially for $E^{*}$. For $-\infty<s<-n$ apply Theorem 3 with
$\varphi(t)=t^{p}$, $k(t)=\pi^{-n-s}-t^{-n-s}$ and $\rho(t)=1$. Then substract
$2\cdot\pi^{-n-s}P_{-n}(E)$ on both sides of the inequality. $\blacksquare$

##  4 Convergence of fractional perimeters as $s\nearrow 1$

We start with a result for subsets of intervals and show that only the
behaviour in a neighbourhood of their boundary points contributes to the
limit:

Lemma 4.1. Let $0<s<1$ and $I\subseteq\mathbb{R}$ be a (possibly unbounded)
closed interval. Suppose that $E=\bigcup_{i=1}^{M}[a_{i},b_{i}]\subseteq I$,
where $M\in\mathbb{N}$ and $a_{1}<b_{1}<a_{2}<\dots<a_{M}<b_{M}$, and that
$I\backslash E=\bigcup_{k=1}^{N}J_{k}$ is the corresponding decomposition into
pairwise disjoint intervals $J_{k}\subseteq\mathbb{R}$. Furthermore, let $l$,
the minimal length of any of the intervals $[a_{i},b_{i}]$ and $J_{k}$, be
greater than 0. Then for any $0<\varepsilon<\frac{l}{2}$

$\lim_{s\nearrow 1}(1-s)\int_{E}\int_{I\backslash
E}\frac{1}{|x-y|^{1+s}}\,\mathrm{d}y\,\mathrm{d}x=\lim_{s\nearrow
1}(1-s)\displaystyle\iint\limits_{F_{\varepsilon}}\frac{1}{|x-y|^{1+s}}\,\mathrm{d}y\,\mathrm{d}x=\mathcal{H}^{0}(\partial
E)$ (4.1)

where $F_{\varepsilon}:=\left\\{(x,y)\in E\times(I\backslash
E):|x-y|<\varepsilon\right\\}$ and $\partial E$ is the boundary of $E$ with
respect to the relative topology on $I$.

Proof. The leftmost limit in (4.1) was evaluated in [15, Lemma 1] for the case
$I=\mathbb{R}$.

For any interval $J_{k}\subseteq I\backslash E$ with endpoints
$-\infty<\alpha<\beta<\infty$ (the case $\alpha=-\infty$ or $\beta=+\infty$
works analogously) we have

$\int_{\alpha}^{\beta}\int_{a_{i}}^{b_{i}}\frac{1}{|x-y|^{1+s}}\,\mathrm{d}y\,\mathrm{d}x=\frac{1}{s(1-s)}[(a_{i}-\alpha)^{1-s}-(a_{i}-\beta)^{1-s}+(b_{i}-\beta)^{1-s}-(b_{i}-\alpha)^{1-s}].$

In the cases that $\beta=a_{i}$ or $\alpha=b_{i}$ we thus get

$\lim_{s\nearrow
1}(1-s)\int_{\alpha}^{\beta}\int_{a_{i}}^{b_{i}}\frac{1}{|x-y|^{1+s}}\,\mathrm{d}y\,\mathrm{d}x=1,$

otherwise this limit is equal to $0$. If $a_{1}=\min I$, then $a_{1}$ does not
lie in the boundary of $E$ relative to $I$ and is not an endpoint of any
interval $J_{k}$, thus

$\lim_{s\nearrow
1}(1-s)\sum_{k=1}^{N}\int_{a_{1}}^{b_{1}}\int_{J_{k}}\frac{1}{|x-y|^{1+s}}\,\mathrm{d}y\,\mathrm{d}x=1,$

since only the boundary point $b_{1}$ contributes to the limit. Otherwise, if
$a_{1}>\min I$, then the above limit is equal to $2$. A similar distinction is
necessary for $b_{M}$ and $\max I$. Summing up over all $i=1,\dots,M$ leads to
the first identity in (4.1).

To see the second identity in (4.1), we only need to evaluate integrals of the
form

$\int_{a}^{a+\varepsilon}\int_{x-\varepsilon}^{a}\frac{1}{|x-y|^{1+s}}\,\mathrm{d}y\,\mathrm{d}x=\int_{b-\varepsilon}^{b}\int_{b}^{x+\varepsilon}\frac{1}{|x-y|^{1+s}}\,\mathrm{d}y\,\mathrm{d}x=\frac{1}{s}\left(\frac{\varepsilon^{1-s}}{1-s}-\varepsilon^{1-s}\right).$
(4.2)

It is easy to see that after multiplying all sides with the factor $(1-s)$,
the right-hand side tends to $1$ as $s\nearrow 1$. Now a similar argument as
before, taking into account the position of $a_{1}$ and $b_{M}$ relative to
$I$, yields the second equality in formula (4.1). $\blacksquare$

As a simple application we get a convergence result for subsets on a curve:

Corollary 4.2. Let $I\subseteq\mathbb{R}$ be a (possibly unbounded) closed
interval and $\gamma:I\to\mathbb{R}^{n}$ be a simple $C^{1}$ curve. If
$E\subseteq I$ is a finite union of closed and pairwise disjoint intervals,
then

$\lim_{s\nearrow
1}(1-s)\int_{\gamma(E)}\int_{\gamma(I)\backslash\gamma(E)}\frac{1}{d_{\gamma}(x,y)^{1+s}}\,\mathrm{d}\mathcal{H}^{1}(y)\,\mathrm{d}\mathcal{H}^{1}(x)=\mathcal{H}^{0}(\partial\gamma(E)),$
(4.3)

where $\partial\gamma(E)$ is the boundary of $\gamma(E)$ relative to
$\gamma(I)$ and $d_{\gamma}(\cdot,\cdot)$ denotes the distance on the curve,
i.e. if $x=\gamma(t_{1})$ and $y=\gamma(t_{2})$, then

$d_{\gamma}(x,y)=\left|\int_{t_{1}}^{t_{2}}|\gamma^{\prime}(t)|\,\mathrm{d}t\right|.$

Proof. Since line integrals do not depend on the parametrization of the curve,
we can assume that $\gamma$ is an arc-length parametrization, i.e.
$|\gamma^{\prime}(t)|=1$ for all $t\in I$. The line integrals in (4.3) can
thus be rewritten as

$\int_{\gamma(E)}\int_{\gamma(I)\backslash\gamma(E)}\frac{1}{d_{\gamma}(x,y)^{1+s}}\,\mathrm{d}\mathcal{H}^{1}(y)\,\mathrm{d}\mathcal{H}^{1}(x)=\int_{E}\int_{I\backslash
E}\frac{1}{|t-u|^{1+s}}\,\mathrm{d}u\,\mathrm{d}t.$

The result then follows from Lemma 4.

$\blacksquare$

In [1, Theorem 1] a higher order kinematic formula on the sphere is proven
from which a spherical Blaschke-Petkantschin formula follows. The direct
statement of the spherical Blaschke-Petkantschin formula with a shorter and
better accessible proof can be found in [12, Lemma 5.3]. In the case of double
integrals the formula reads as follows.

Theorem 4.3 (spherical Blaschke-Petkantschin formula).  Let
$f:\mathbb{S}^{n}\times\mathbb{S}^{n}\to[0,\infty)$ be a measurable function.
Then

$\displaystyle\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}f(x,y)$
$\displaystyle\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)$
$\displaystyle=c_{n}\int_{G(n+1,2)}\int_{\mathbb{S}^{n}\,\cap\,L}\int_{\mathbb{S}^{n}\,\cap\,L}f(x,y)\nabla_{2}^{n-1}(x,y)\,\mathrm{d}\mathcal{H}^{1}(y)\,\mathrm{d}\mathcal{H}^{1}(x)\,\mathrm{d}L,$
(4.4)

where $c_{n}:=\frac{\omega_{n+1}\omega_{n}}{\omega_{1}\omega_{2}}$ and
$\nabla_{2}(x,y)$ denotes the area of the parallelogram spanned by $x$ and
$y$.

Note that for points $x,y\in\mathbb{S}^{n}$ on the sphere

$\nabla_{2}(x,y)=\sqrt{1-(x\cdot y)^{2}}=\sin\sphericalangle(x,y),$

where $\sphericalangle(x,y)$ is the unorientated angle between $x$ and $y$.

The next result, a spherical Crofton formula, identifies the average number of
intersections of a polyconvex subset $E\subseteq\mathbb{S}^{n}$ with great
circles as its perimeter:

Theorem 4.4 (spherical Crofton formula).  Let $E\subseteq\mathbb{S}^{n}$ be an
$n$-dimensional polyconvex subset of $\mathbb{S}^{n}$. Then

$\int_{G(n+1,2)}\mathcal{H}^{0}(\partial
E\,\cap\,L)\,\mathrm{d}L=\frac{2}{\omega_{n}}\mathcal{H}^{n-1}(\partial E).$

Proof. We rewrite the spherical Crofton formula for curvature measures,

$\int_{G(n+1,2)}\varphi_{0}(E\,\cap\,L,\mathbb{S}^{n}\,\cap\,L)\,\mathrm{d}L=\varphi_{n-1}(E,\mathbb{S}^{n}),$
(4.5)

see [22, p. 261], in terms of Hausdorff measures as follows:

For spherical convex polytopes $P\subseteq\mathbb{S}^{n}$ and
$m\in\left\\{0,\dots,n-1\right\\}$, we have the formula

$\varphi_{m}(P,A)=\frac{1}{\omega_{m+1}\omega_{n-m}}\sum_{F\in\mathcal{F}_{m}(P)}\int_{F}\int_{N(P,F)}\mathds{1}_{A\times\mathbb{S}^{n}}(x,u)\,\mathrm{d}\mathcal{H}^{n-m-1}(u)\,\mathrm{d}\mathcal{H}^{m}(x),$

where $\mathcal{F}_{m}(P)$ denotes the set of all $m$-dimensional faces of $P$
and $N(P,F)$ is the normal cone to $P$ for $F$ (see [22, Theorem 6.5.1]).

Since $E\,\cap\,L$ is a finite union of disjoint arcs and
$\varphi_{0}(\cdot,A)$ is a valuation, it suffices to calculate
$\varphi_{0}(P,S)$, where $P$ is an arc on the great sphere
$S=\mathbb{S}^{n}\,\cap\,L$. The set of $0$-dimensional faces
$\mathcal{F}_{0}(P)=\left\\{p_{1},p_{2}\right\\}$ consists of the endpoints of
the arc, and for every normal vector $u\in N(P,p_{i}),i=1,2$ we have
$\mathds{1}_{S\times\mathbb{S}^{n}}(p_{i},u)=1$, so it remains to compute
$\mathcal{H}^{n-1}(N(P,p_{i}))$.

We can think of $P$ as an intersection of the great sphere $S$ with two
hemispheres, with normal vectors lying in the same plane as $S$ each.
Therefore, the polar body $P^{\circ}$ of $P$ is again an intersection of two
hemispheres, and since for $i=1,2$ the normal cone $N(P,p_{i})$ consists of
one of the bounding $(n-1)$-dimensional hemispheres, we have
$\mathcal{H}^{n-1}(N(P,p_{i}))=\frac{\omega_{n}}{2}$.

In conclusion, for any polyconvex set $E\subseteq\mathbb{S}^{n}$ and any plane
$L\in G(n+1,2)$, counting the components of $E\,\cap\,L$ yields

$\varphi_{0}(E\,\cap\,L,\mathbb{S}^{n}\,\cap\,L)=\frac{\mathcal{H}^{0}(\partial
E\,\cap\,L)}{2\omega_{1}}.$

Regarding the right-hand side of (4.5), [11, Satz 4.4.3] identifies the
curvature measure $\varphi_{n-1}(E,\mathbb{S}^{n})$ as
$\frac{1}{2\omega_{n}}\mathcal{H}^{n-1}(\partial E)$. $\blacksquare$

Now we turn to the proof of our first convergence result:

Theorem 4.5.  Let $E$ be a polyconvex subset of $\mathbb{S}^{n}$. Then

$\lim_{s\nearrow
1}(1-s)\int_{E}\int_{E^{c}}\frac{1}{d(x,y)^{n+s}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)=\frac{\omega_{n+1}}{\omega_{2}}\mathcal{H}^{n-1}(\partial
E).$ (4.6)

Proof. If $E$ is an $\mathcal{H}^{n}$-nullset, then both sides of (4.6) are
equal to 0, so suppose $\mathcal{H}^{n}(E)>0$.

We apply the spherical Blaschke-Petkantschin formula (4.4) to the left-hand
side of (4.6) which results in

$\displaystyle(1-s)$
$\displaystyle\int_{E}\int_{E^{c}}\frac{1}{d(x,y)^{n+s}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)$
$\displaystyle=c_{n}(1-s)\int_{G(n+1,2)}\int_{E\,\cap\,L}\int_{E^{c}\,\cap\,L}\frac{\nabla_{2}^{n-1}(x,y)}{d(x,y)^{n+s}}\,\mathrm{d}\mathcal{H}^{1}(y)\,\mathrm{d}\mathcal{H}^{1}(x)\,\mathrm{d}L$
$\displaystyle=c_{n}(1-s)\int_{G(n+1,2)}\int_{A_{L}}\int_{[0,2\pi]\backslash
A_{L}}\frac{\sin^{n-1}(\delta(\varphi,\psi))}{\delta(\varphi,\psi)^{n+s}}\,\mathrm{d}\psi\,\mathrm{d}\varphi\,\mathrm{d}L,$
(4.7)

where in the last step we introduced an arc-length parametrization
$\gamma_{L}:[0,2\pi]\to\mathbb{S}^{n}\,\cap\,L$ of the great circle
$\mathbb{S}^{n}\,\cap\,L$ such that
$d(\gamma_{L}(\varphi),\gamma_{L}(\psi))=\delta(\varphi,\psi)$, where

$\delta(\varphi,\psi)=\begin{cases}|\varphi-\psi|,&\text{if
}|\varphi-\psi|\leq\pi,\\\ 2\pi-|\varphi-\psi|,&\text{else},\end{cases}$

and put $A_{L}:=\gamma_{L}^{-1}(E\,\cap\,L)$. We only consider the case
$E\,\cap\,L\neq\mathbb{S}^{n}\,\cap\,L$ (otherwise the inner integrals equal
$0$), such that $E^{c}\,\cap\,L$ is the finite union of nonempty open circular
arcs and choose the parametrization $\gamma_{L}$ such that
$\gamma_{L}(0)=\gamma_{L}(2\pi)$ lies in one of the open arcs.

Taylor expansion of the nominator in the integrand yields

$\frac{\sin^{n-1}(\delta(\varphi,\psi))}{\delta(\varphi,\psi)^{n+s}}=\frac{\delta(\varphi,\psi)^{n-1}(1+O(\delta(\varphi,\psi)^{2}))^{n-1}}{\delta(\varphi,\psi)^{n+s}}=\frac{1+r(\delta(\varphi,\psi))}{\delta(\varphi,\psi)^{1+s}},$

where for the remainder $r(t)$ there exists $\varepsilon>0$ and a constant
$C>0$ such that $|r(t)|\leq Ct^{2}$ as long as $t<\varepsilon$. Now divide the
domain of integration into

$\displaystyle M_{<\varepsilon}$ $\displaystyle:=\left\\{(\varphi,\psi)\in
A_{L}\times([0,2\pi]\backslash
A_{L}):\delta(\varphi,\psi)<\varepsilon\right\\},\ \text{and}$ $\displaystyle
M_{\geq\varepsilon}$ $\displaystyle:=\left\\{(\varphi,\psi)\in
A_{L}\times([0,2\pi]\backslash
A_{L}):\delta(\varphi,\psi)\geq\varepsilon\right\\}.$

By our choice of parametrization, if $\varepsilon$ is small enough, then all
pairs of the form $(\varphi,0)$ or $(\varphi,2\pi),\varphi\in A_{L},$ do not
lie in $M_{<\varepsilon}$ such that $\delta(\varphi,\psi)=|\varphi-\psi|$ in
$M_{<\varepsilon}$. Since

$\displaystyle\Bigg{|}\iint\limits_{M_{<\varepsilon}}\frac{r(|\varphi-\psi|)}{|\varphi-\psi|^{1+s}}\,\mathrm{d}\psi\,\mathrm{d}\varphi\Bigg{|}$
$\displaystyle\leq
C\iint\limits_{M_{<\varepsilon}}|\varphi-\psi|^{1-s}\,\mathrm{d}\psi\,\mathrm{d}\varphi\leq
C\mathcal{H}^{2}(M_{<\varepsilon})\varepsilon^{1-s}\leq(2\pi)^{2}C\varepsilon^{1-s}$

by Lemma 4 we get

$\lim_{s\nearrow 1}\
(1-s)\iint\limits_{M_{<\varepsilon}}\frac{\sin^{n-1}|\varphi-\psi|}{|\varphi-\psi|^{n+s}}\,\mathrm{d}\psi\,\mathrm{d}\varphi=\mathcal{H}^{0}(\partial
A_{L}).$

Since the integrand has no singularities in $M_{\geq\varepsilon}$ by dominated
convergence we readily have

$\lim_{s\nearrow 1}\
(1-s)\iint\limits_{M_{\geq\varepsilon}}\frac{\sin^{n-1}(\delta(\varphi,\psi))}{\delta(\varphi,\psi)^{n+s}}\,\mathrm{d}\psi\,\mathrm{d}\varphi=0.$

By the finiteness of the measure on $G(n+1,2)$, we can exchange limit and
integration over $G(n+1,2)$ in (4.7) to eventually obtain

$\displaystyle\lim_{s\nearrow 1}(1-s)\int_{E}\int_{\mathbb{S}^{n}\backslash
E}\frac{1}{d(x,y)^{n+s}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)$
$\displaystyle=c_{n}\int_{G(n+1,2)}\mathcal{H}^{0}(\partial
E\,\cap\,L)\,\mathrm{d}L$
$\displaystyle=\frac{\omega_{n+1}}{\omega_{2}}\mathcal{H}^{n-1}(\partial E),$

where for the last equality we applied the spherical Crofton formula, Theorem
4. $\blacksquare$

##  5 Convergence of fractional perimeters as $s\searrow-\infty$

In view of the result due to Maz’ya & Shaposhnikova ([21], see (1.6)) it would
be natural to consider

$\displaystyle\lim_{s\searrow
0}s\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}\frac{|f(x)-f(y)|}{d(x,y)^{n+s}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x).$
(5.1)

However, the following considerations show that we do not get anything similar
to the integral of $f$. In the Euclidean setting we already observe for the
fractional $0$-seminorm of a smooth function $f$ with compact support in an
open Euclidean ball $B$ that

$\int_{B}\int_{B}\frac{|f(x)-f(y)|}{|x-y|^{n}}\,\mathrm{d}y\,\mathrm{d}x\leq\max_{\xi\in
B}|\nabla
f(\xi)|\int_{B}\int_{B}\frac{1}{|x-y|^{n-1}}\,\mathrm{d}y\,\mathrm{d}x<\infty.$

For the sphere, the finiteness of fractional seminorms for smooth functions
then follows from introducing suitable coordinates and using the argument
above. Hence, (5.1) is always $0$ for smooth functions.

Denote by $\tilde{d}(x,y)=\frac{d(x,y)}{\pi}\in[0,1]$ the normalized geodesic
distance between two points $x,y\in\mathbb{S}^{n}$. Furthermore we put
$t:=-s$.

Lemma 5.1. Let $1\leq p<\infty$. For every $x\in\mathbb{S}^{n}$ and $\delta>0$

$\lim_{t\to\infty}t^{n}\int_{C(-x,\delta)}\tilde{d}(x,y)^{-n+tp}\,\mathrm{d}\mathcal{H}^{n}(y)=\frac{\mathcal{H}^{n-1}(\mathbb{S}^{n-1})\pi^{n}(n-1)!}{p^{n}},$

where $C(-x,\delta)=\left\\{y\in\mathbb{S}^{n}:d(-x,y)<\delta\right\\}$ is the
open spherical cap around $-x$ with radius $\delta$ (if $\delta>\pi$, then
$C(-x,\delta)=\mathbb{S}^{n})$.

Proof. For $\varepsilon>0$ there exist $\delta_{0}\in(0,\delta)$ and a normal
coordinate chart $\varphi:C(-x,\delta_{0})\to B_{\delta_{0}}^{n}$ such that

$\displaystyle\tilde{d}(x,y)=1-\frac{|\varphi(y)|}{\pi},\quad\text{ and}$
$\displaystyle 1-\varepsilon\leq\sqrt{\det(g_{ij}(y))}\leq 1+\varepsilon$

for all $y\in C(-x,\delta_{0})$. Observe that

$t^{n}\int_{C(-x,\delta)\backslash
C(-x,\delta_{0})}\tilde{d}(x,y)^{-n+tp}\,\mathrm{d}\mathcal{H}^{n}(y)\leq\mathcal{H}^{n}(\mathbb{S}^{n})t^{n}\left(1-\frac{\delta_{0}}{\pi}\right)^{-n+tp}\to
0$

as $t\to\infty$, thus

$\lim_{t\to\infty}t^{n}\int_{C(-x,\delta)}\tilde{d}(x,y)^{-n+tp}\,\mathrm{d}\mathcal{H}^{n}(y)=\lim_{t\to\infty}t^{n}\int_{C(-x,\delta_{0})}\tilde{d}(x,y)^{-n+tp}\,\mathrm{d}\mathcal{H}^{n}(y).$

By our choice of coordinates we have

$\displaystyle
t^{n}\int_{C(-x,\delta_{0})}\tilde{d}(x,y)^{-n+tp}\,\mathrm{d}\mathcal{H}^{n}(y)$
$\displaystyle\leq(1+\varepsilon)t^{n}\int_{B_{\delta_{0}}^{n}}\left(1-\frac{|\varphi(y)|}{\pi}\right)^{-n+tp}\,\mathrm{d}y$
$\displaystyle=(1+\varepsilon)\mathcal{H}^{n-1}(\mathbb{S}^{n-1})t^{n}\int_{0}^{\delta_{0}}\left(1-\frac{r}{\pi}\right)^{-n+tp}r^{n-1}\,\mathrm{d}r$
$\displaystyle=(1+\varepsilon)\mathcal{H}^{n-1}(\mathbb{S}^{n-1})\pi^{n}t^{n}\int_{0}^{\delta_{0}/\pi}(1-u)^{-n+tp}u^{n-1}\,\mathrm{d}u$

where we introduced the substitution $u=\frac{r}{\pi}$ in the last step. The
last integral is equal to $B_{\frac{\delta_{0}}{\pi}}(n,-n+tp+1)$, where
$B_{T}(a,b)=\int_{0}^{T}u^{a-1}(1-u)^{b-1}\,\mathrm{d}u$ is the incomplete
Beta function. Note that

$\lim_{t\to\infty}t^{n}B_{\frac{\delta_{0}}{\pi}}(n,-n+tp+1)=\lim_{t\to\infty}t^{n}B(n,-n+tp+1)$

where $B(a,b)=\int_{0}^{1}u^{a-1}(1-u)^{b-1}\,\mathrm{d}u$ is the (complete)
Beta function since

$\displaystyle
t^{n}\int_{\frac{\delta_{0}}{\pi}}^{1}u^{n-1}(1-u)^{-n+tp}\,\mathrm{d}u\leq
t^{n}\left(1-\frac{\delta_{0}}{\pi}\right)^{-n+tp+1}\to 0,$

as $t\to\infty$. Thus it suffices to determine the value of
$\displaystyle\lim_{t\to\infty}t^{n}B(n,-n+tp+1)$. The identity
$B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$ together with Stirling’s
formula

$\Gamma(x)=\sqrt{\frac{2\pi}{x}}\left(\frac{x}{e}\right)^{x}e^{\mu(x)},\quad\text{where
}0<\mu(x)<\frac{1}{12x}$

can be used to deduce

$\displaystyle\lim_{t\to\infty}t^{n}B(n,-n+tp+1)$
$\displaystyle=\Gamma(n)\lim_{t\to\infty}\sqrt{\frac{tp+1}{tp+1-n}}e^{n}\left(1-\frac{n}{tp+1}\right)^{tp+1}\frac{t^{n}}{(tp+1-n)^{n}}\frac{e^{\mu(-n+tp+1)}}{e^{\mu(tp+1)}}$
$\displaystyle=\frac{(n-1)!}{p^{n}}.$

Similarly, one can prove that the reverse inequality

$\lim_{t\to\infty}t^{n}\int_{C(-x,\delta_{0})}\tilde{d}(x,y)^{-n+tp}\,\mathrm{d}\mathcal{H}^{n}(y)\geq(1-\varepsilon)\frac{\mathcal{H}^{n-1}(\mathbb{S}^{n-1})\pi^{n}(n-1)!}{p^{n}}$

holds. Thus the result follows from letting $\varepsilon\to 0$. $\blacksquare$

The next theorem shows the convergence of fractional seminorms as
$s\to-\infty$. Note that the limit measures the reflection symmetry of a
function in the $L_{p}$-sense, i.e. the integral in the right-hand side of
(5.2) is $0$ precisely for functions which are even almost everywhere and it
is equal to $(2\|f\|_{p})^{p}$ precisely for functions which are odd almost
everywhere.

Theorem 5.2.  Let $1\leq p<\infty$ and $f\in L_{p}(\mathbb{S}^{n})$. Then

$\lim_{t\to\infty}t^{n}\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}\frac{|f(x)-f(y)|^{p}}{\tilde{d}(x,y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)=c_{n,p}\int_{\mathbb{S}^{n}}|f(x)-f(-x)|^{p}\,\mathrm{d}\mathcal{H}^{n}(x)$
(5.2)

where
$c_{n,p}=\frac{\mathcal{H}^{n-1}(\mathbb{S}^{n-1})\pi^{n}(n-1)!}{p^{n}}$.

Proof. We split the proof into two steps. In the first step we show (5.2) for
continuous functions and use a density argument in the second step to extend
the formula to general $L_{p}$-functions.

Step 1: Proof for continuous functions

Let $g$ be a continuous function on $\mathbb{S}^{n}$. First, note that by the
dominated convergence theorem and Lemma 5

$\lim_{t\to\infty}t^{n}\int_{\mathbb{S}^{n}}\left(\int_{\mathbb{S}^{n}}\frac{|g(x)-g(-x)|^{p}}{\tilde{d}(x,y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\right)\,\mathrm{d}\mathcal{H}^{n}(x)=c_{n,p}\int_{\mathbb{S}^{n}}|g(x)-g(-x)|^{p}\,\mathrm{d}\mathcal{H}^{n}(x)$
(5.3)

since the integrand for $x$-integration is dominated by
$(2\|g\|_{\infty})^{p}C$ with a constant $C>0$ for sufficiently large $t$.
Moreover, we have for every $0<\delta<\pi$ that

$\displaystyle\lim_{t\to\infty}t^{n}\int_{\mathbb{S}^{n}}$
$\displaystyle\int_{\mathbb{S}^{n}\backslash
C(-x,\delta)}\frac{|g(x)-g(-x)|^{p}}{\tilde{d}(x,y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)$
$\displaystyle=\lim_{t\to\infty}t^{n}\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}\backslash
C(-x,\delta)}\frac{|g(x)-g(y)|^{p}}{\tilde{d}(x,y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)=0,$
(5.4)

i.e. the inner integrals concentrate on the point $-x$ in the limit.

Now let $\varepsilon>0$ and choose $\delta>0$ be such that
$|g(y)-g(-x)|<\varepsilon$ whenever $x\in\mathbb{S}^{n}$ and $y\in
C(-x,\delta)$. Using Taylor’s formula for the case $p>1$ and the reverse
triangle inequality for the case $p=1$ we rewrite

$|g(x)-g(y)|^{p}=|g(x)-g(-x)|^{p}+r(x,y)$

where the remainder term satisfies $|r(x,y)|\leq c\cdot\varepsilon$ for all
$y\in C(-x,\delta)$ with a constant $c$ independent of $x$ and $y$. From this
and (5.4) it follows that

$\displaystyle\limsup_{t\to\infty}\left|\,t^{n}\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}\right.$
$\displaystyle\left.\frac{|g(x)-g(y)|^{p}}{\tilde{d}(x,y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)-t^{n}\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}\frac{|g(x)-g(-x)|^{p}}{\tilde{d}(x,y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)\right|$
$\displaystyle\leq
c\cdot\varepsilon\limsup_{t\to\infty}t^{n}\int_{\mathbb{S}^{n}}\int_{C(-x,\delta)}\tilde{d}(x,y)^{-n+tp}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x).$

Now formula (5.2) for continuous functions follows from the arbitrariness of
$\varepsilon>0$ and (5.3).

Step 2: Proof for general $f\in L_{p}$

Let $f\in L_{p}(\mathbb{S}^{n})$ and define
$\mathcal{S}f:\mathbb{S}^{n}\to\mathbb{R}$ by
$\mathcal{S}f(x):=c_{n,p}^{\frac{1}{p}}|f(x)-f(-x)|$, and for $t\in\mathbb{R}$
the function $F_{t}:\mathbb{S}^{n}\to\mathbb{R}$ by

$F_{t}(x):=\left(t^{n}\int_{\mathbb{S}^{n}}\frac{|f(x)-f(y)|^{p}}{\tilde{d}(x,y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\right)^{\frac{1}{p}}.$

We show that $F_{t}\stackrel{{\scriptstyle t\to\infty}}{{\to}}\mathcal{S}f$ in
$L_{p}(\mathbb{S}^{n})$ which implies that
$\displaystyle\lim_{t\to\infty}\|F_{t}\|_{p}^{p}=\|\mathcal{S}f\|_{p}^{p}$ and
thus formula (5.2). By density, for each $\varepsilon>0$ there exists a
continuous function $g$ on $\mathbb{S}^{n}$ such that
$\|f-g\|_{p}<\varepsilon$. With $\mathcal{S}g$ and $G_{t}$ defined as above,
we have

$\displaystyle\|F_{t}-\mathcal{S}f\|_{p}\leq\|F_{t}-G_{t}\|_{p}+\|G_{t}-\mathcal{S}g\|_{p}+\|\mathcal{S}g-\mathcal{S}f\|_{p}.$

By step 1, the summand $\|G_{t}-\mathcal{S}g\|_{p}$ tends to 0 as $t$ goes to
infinity. Moreover, by rotation invariance of the Hausdorff measure,

$\displaystyle\|\mathcal{S}g-\mathcal{S}f\|_{p}=c_{n,p}^{\frac{1}{p}}\left(\int_{\mathbb{S}^{n}}\bigg{|}|g(x)-g(-x)|-|f(x)-f(-x)|\bigg{|}^{p}\,\mathrm{d}\mathcal{H}^{n}(x)\right)^{\frac{1}{p}}\leq
2c_{n,p}^{\frac{1}{p}}\|f-g\|_{p}<2c_{n,p}^{\frac{1}{p}}\cdot\varepsilon.$

For the remaining summand, we first observe that

$F_{t}(x)=t^{\frac{n}{p}}\left\|\frac{f(x)-f(\cdot)}{\tilde{d}(x,\cdot)^{\frac{n}{p}-t}}\right\|_{p}$

and by the triangle inequality for $L_{p}$-norms

$\displaystyle\|F_{t}-G_{t}\|_{p}$
$\displaystyle=\left(\int_{\mathbb{S}^{n}}|F_{t}(x)-G_{t}(x)|^{p}\,\mathrm{d}\mathcal{H}^{n}(x)\right)^{\frac{1}{p}}\leq
t^{\frac{n}{p}}\left(\int_{\mathbb{S}^{n}}\left|\left\|\frac{f(x)-g(x)}{\tilde{d}(x,\cdot)^{\frac{n}{p}-t}}\right\|_{p}+\left\|\frac{f(\cdot)-g(\cdot)}{\tilde{d}(x,\cdot)^{\frac{n}{p}-t}}\right\|_{p}\right|^{p}\,\mathrm{d}\mathcal{H}^{n}(x)\right)^{\frac{1}{p}}$
$\displaystyle\leq
2\left(t^{n}\int_{\mathbb{S}^{n}}\int_{\mathbb{S}^{n}}\frac{|f(x)-g(x)|^{p}}{\tilde{d}(x,y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)\right)^{\frac{1}{p}}=2\left(\int_{\mathbb{S}^{n}}\frac{t^{n}}{\tilde{d}(e_{1},y)^{n-tp}}\,\mathrm{d}\mathcal{H}^{n}(y)\right)^{\frac{1}{p}}\|f-g\|_{p}$
$\displaystyle<\text{const}\cdot\varepsilon,$

where $e_{1}=(1,0,\dots,0)$ and the constant does not depend on $t$.
$\blacksquare$

Corollary 5.3. Let $E\subseteq\mathbb{S}^{n}$ be a Borel set. Then,

$\lim_{t\to\infty}t^{n}\int_{E}\int_{E^{c}}\frac{1}{\tilde{d}(x,y)^{n-t}}\,\mathrm{d}\mathcal{H}^{n}(y)\,\mathrm{d}\mathcal{H}^{n}(x)=c_{n,1}\mathcal{H}^{n}((-E)\,\cap\,(E^{c})),$

where $c_{n,1}=\mathcal{H}^{n-1}(\mathbb{S}^{n-1})\pi^{n}(n-1)!$.

Proof. The statement follows from Theorem 5 with $f=\mathds{1}_{E}$ and $p=1$.
$\blacksquare$

### Acknowledgement

The authors would like to thank Monika Ludwig for helpful comments and
suggestions during the preparation of this paper.

## References

  * [1] E. Arbeiter and M. Zähle.  Kinematic relations for Hausdorff moment measures in spherical spaces.  Math. Nachr., 153:333–348, 1991.
  * [2] W. Beckner.  Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^{n}$.  Proc. Nat. Acad. Sci. U.S.A., 89(11):4816–4819, 1992.
  * [3] F. Besau and E. Werner.  The spherical convex floating body.  Adv. Math., 301:867–901, 2016.
  * [4] C. Borell.  The Brunn-Minkowski inequality in Gauss space.  Invent. Math., 30(2):207–216, 1975.
  * [5] J. Bourgain, H. Brézis, and P. Mironescu.  Another look at Sobolev spaces.  In Optimal control and partial differential equations, pages 439–455. IOS, Amsterdam, 2001.
  * [6] J. Bourgain, H. Brézis, and P. Mironescu.  Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications.  J. Anal. Math., 87:77–101, 2002.
  * [7] L. Caffarelli, J.-M. Roquejoffre, and O. Savin.  Nonlocal minimal surfaces.  Comm. Pure Appl. Math., 63(9):1111–1144, 2010.
  * [8] J. Dávila.  On an open question about functions of bounded variation.  Calc. Var. Partial Differential Equations, 15(4):519–527, 2002\.
  * [9] E. De Giorgi.  Definizione ed espressione analitica del perimetro di un insieme.  Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 14:390–393, 1953.
  * [10] R. L. Frank and R. Seiringer.  Non-linear ground state representations and sharp Hardy inequalities.  J. Funct. Anal., 255(12):3407–3430, 2008.
  * [11] S. Glasauer.  Integralgeometrie konvexer Körper im sphärischen Raum.  Dissertation, Universiät Freiburg, 1995.
  * [12] D. Hug and C. Thäle.  Splitting tessellations in spherical spaces.  Electron. J. Probab., 24:60 pp., 2019.
  * [13] A. Kreuml.  The anisotropic fractional isoperimetric problem with respect to unconditional unit balls.  Commun. Pure Appl. Anal., in press.
  * [14] A. Kreuml and O. Mordhorst.  Fractional Sobolev norms and BV functions on manifolds.  Nonlinear Anal., 187:450–466, 2019.
  * [15] M. Ludwig.  Anisotropic fractional perimeters.  J. Differential Geom., 96(1):77–93, 2014.
  * [16] M. Ludwig.  Anisotropic fractional Sobolev norms.  Adv. Math., 252:150–157, 2014.
  * [17] E. Lutwak.  The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas.  Adv. Math., 118(2):244–294, 1996.
  * [18] E. Lutwak, D. Yang, and G. Zhang.  $L_{p}$ affine isoperimetric inequalities.  J. Differential Geom., 56(1):111–132, 2000.
  * [19] E. Lutwak, D. Yang, and G. Zhang.  Sharp affine $L_{p}$ Sobolev inequalities.  J. Differential Geom., 62(1):17–38, 2002.
  * [20] F. Maggi.  Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory.  Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2012.
  * [21] V. Maz’ya and T. Shaposhnikova.  On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces.  J. Funct. Anal., 195(2):230–238, 2002.
  * [22] R. Schneider and W. Weil.  Stochastic and Integral Geometry.  Probability and its Applications (New York). Springer-Verlag, Berlin, 2008\.
  * [23] V. N. Sudakov and B. S. Tsirelson.  Extremal properties of half-spaces for spherically invariant measures.  Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 41:14–24, 165, 1974.  Problems in the theory of probability distributions, II.
  * [24] E. Werner.  The $p$-affine surface area and geometric interpretations.  Number 70, part II, pages 367–382. 2002.  IV International Conference in “Stochastic Geometry, Convex Bodies, Empirical Measures $\&$ Applications to Engineering Science”, Vol. II (Tropea, 2001).
  * [25] E. Werner and D. Ye.  New $L_{p}$ affine isoperimetric inequalities.  Adv. Math., 218(3):762–780, 2008.
  * [26] J. A. Wieacker.  Translative Poincaré formulae for Hausdorff rectifiable sets.  Geom. Dedicata, 16(2):231–248, 1984.

Andreas Kreuml

Institut für Diskrete Mathematik und Geometrie
Technische Universität Wien
Wiedner Hauptstraße 8-10/1046
1040 Vienna, Austria
E-mail<EMAIL_ADDRESS>
Olaf Mordhorst

Institut für Mathematik
Goethe-Universität Frankfurt am Main
Robert-Mayer-Straße 10
60325 Frankfurt am Main, Germany
E-mail<EMAIL_ADDRESS>