A singular integral identity for surface measure

Abstract

We prove that the integral of a certain Riesz-type kernel over $(n-1)$-rectifiable sets in ${\mathbb{R}}^n$ is constant, from which a formula for surface measure immediately follows. Geometric interpretations are given, and the solution to a geometric variational problem characterizing convex domains follows as a corollary, strengthening a recent inequality of Steinerberger.

arXiv comment10 pages, 4 figures; Correction to the proof of the Corollary; To appear in the Journal of Geometric Analysis