Fractional perimeters on the sphere

Abstract

<p style='text-indent:20px;'>This note treats several problems for the fractional perimeter or <inline-formula><tex-math id="M1">\begin{document}$ s $\end{document}</tex-math></inline-formula>-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as <inline-formula><tex-math id="M2">\begin{document}$ s \nearrow 1 $\end{document}</tex-math></inline-formula> is proven. It is shown that their limit as <inline-formula><tex-math id="M3">\begin{document}$ s \searrow -\infty $\end{document}</tex-math></inline-formula> can be expressed in terms of the volume.</p>