Riesz capacity: monotonicity, continuity, diameter and volume

TL;DR

This paper investigates the properties of Riesz capacity, demonstrating its monotonicity, continuity, and how it relates to geometric measures like diameter and volume as the kernel exponent varies.

Contribution

It establishes new monotonicity and continuity results for Riesz capacity with respect to the kernel exponent, linking capacity limits to geometric set properties.

Findings

Capacity is monotonic in the kernel exponent p.

Limits of capacity as p approaches endpoints recover set diameter and volume.

Capacity exhibits left- and right-continuity under certain conditions.

Abstract

Properties of Riesz capacity are developed with respect to the kernel exponent $p \in (-\infty,n)$, namely that capacity is monotonic as a function of $p$, that its endpoint limits recover the diameter and volume of the set, and that capacity is left-continuous with respect to $p$ and is right-continuous provided (when $p \geq 0$) that an additional hypothesis holds. Left and right continuity properties of the equilibrium measure are obtained too.