A singular integral identity for surface measure (Q6067787)

Let \(\Sigma\) be an \((n-1)\)-rectifiable set in \(\mathbb{R}^n\) and \(\nu\) a measurable unit normal vector field on \(\Sigma\) such that the pair \((\Sigma, \nu)\) satisfies the so called \textit{orientation cancellation condition}. The following identity \[ \int_\Sigma\frac{\langle x-y,\nu(y)\rangle\langle y-x,\nu(x)\rangle}{\|x-y\|^{n+1}}\,d\mathcal{H}^{n-1}(y)=\mathcal{L}^{n-1}(B(0, 1)) \] is proved for \({\mathcal{H}^{n-1}}\)-a.e. \(x\in\Sigma.\) This result implies Steinerberger's recent inequality [\textit{S. Steinerberger}, ``An inequality characterizing convex domains'', Preprint, \url{arXiv:2209.14153}] under a milder regularity condition for domain.