A singular integral identity for surface measure

TL;DR

The paper proves a universal singular integral identity for $n-1$-rectifiable sets endowed with a normal field under the orientation cancellation condition (OCC). It leverages a radial projection and the coarea formula to show that the signed kernel integral equals the constant $\alpha_{n-1}$ for a.e. points, yielding a direct formula for the surface measure and strengthening Steinerberger's inequality by showing equality characterizes convex domains. The OCC generalizes orientability beyond smooth boundaries and applies to reduced boundaries of finite-perimeter sets, enabling a nonlocal isoperimetric-type statement. This work links geometric measure theory with nonlocal kernels and convexity via a variational perspective on a broad class of hypersurfaces.

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.

Figures (4)

• Figure 1: For a $C^1$-bounded convex region $\Omega$, the line segment between any two points $x,y \in \partial\Omega$ is such that $\langle x-y, \nu(y) \rangle \langle x-y, \nu(x) \rangle \leq 0$, and this quantity vanishes quickly as $\| x-y \| \to 0$. If $\Omega$ is not convex, then $\langle x-y, \nu(y) \rangle \langle x-y, \nu(x) \rangle$ can be positive and there may be nearby points $x,y \in \partial\Omega$ such that $\langle x-y, \nu(y) \rangle \langle x-y, \nu(x) \rangle$ is not small relative to $\| x-y \|$.
• Figure 2: The normal vectors to this immersed submanifold are oriented in such a way that the signs of the angles formed with any line sum to $0$. This motivates the definition of the "orientation cancellation condition."
• Figure 3: The proof of the Theorem formalizes the following idea: if $\Sigma$ (depicted here as the boundary of a $C^1$-bounded region) is partitioned into double cones with vertex at $x$, then each piece of $\Sigma$ that slices the double cone contributes approximately the same mass to the integral in Equation \ref{['eq:main-1']}, up to a sign. (The weight factor $\langle x-y, \nu(y) \rangle/\| x-y \|^n$ is chosen precisely to guarantee this.) The singleton $\{ x \}$ has no mass, so the OCC implies that the contribution to the integral from this double cone is approximately the area of a single slice of the cone that is unit distance from $x$ and orthogonal to the axis of the cone.
• Figure 4: For almost all $x \in \partial^*\space E$ and $\omega \in \mathbb{S}^{n-1}$, the arrangement of the normal vectors to $\partial^*\space E$ along $L_{x,\omega}$ is the "obvious" one depicted here. This is the thrust of Lemma \ref{['lem:finite-perimeter']} and the reason Equation \ref{['eq:angles']} implies convexity in the proof of the Corollary.

Theorems & Definitions (5)

• Proposition 1
• Theorem 1
• Corollary 1
• Lemma 1
• Lemma 2