Sharp stability for the Riesz potential

TL;DR

This paper proves that the ball is the most stable shape for maximizing the Riesz potential among sets of fixed volume, with a sharp stability exponent, across all dimensions and relevant powers.

Contribution

It establishes the sharp stability of the ball as the maximizer of the Riesz potential for all dimensions and powers within specified ranges.

Findings

Sharp stability exponent of 1/2 proven

Stability holds for all dimensions N≥2

Applicable for all powers 1<α<N

Abstract

In this paper we show the stability of the ball as maximizer of the Riesz potential among sets of given volume. The stability is proved with sharp exponent $1/2$, and is valid for any dimension $N\geq 2$ and any power $1<\alpha<N$.