Riesz capacity: monotonicity, continuity, diameter and volume

Abstract

Properties of Riesz capacity are developed with respect to the kernel exponent \(p \in (-\infty ,n)\), namely that capacity is strictly 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 \(p \ge 0\)) that an additional hypothesis holds. Left and right continuity properties of the equilibrium measure are obtained too.