Characterization of balls through optimal concavity for potential functions

TL;DR

This paper proves that convex domains with a specific concavity property in their p-capacitary potential function must be spherical, characterizing balls through potential function concavity.

Contribution

It establishes a new geometric characterization of balls based on the concavity of their p-capacitary potential functions.

Findings

Convex domains with a certain potential function concavity are necessarily balls.

The result links potential theory and geometric shape characterization.

Provides a new criterion for identifying spherical domains in convex analysis.

Abstract

Let $p\in(1,n)$. If $\Omega$ is a convex domain in $\rn$ whose $p$-capacitary potential function $u$ is $(1-p)/(n-p)$-concave (i.e. $u^{(1-p)/(n-p)}$ is convex), then $\Omega$ is a ball.