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 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.