A singular integral identity for surface measure

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.