An inequality characterizing convex domains

TL;DR

This paper establishes a new inequality involving boundary normals and distances in smooth domains, characterizing convexity through an integral condition that is tight only for convex domains.

Contribution

The authors prove a novel integral inequality involving boundary normals and distances, providing a new characterization of convex domains among smooth bounded domains.

Findings

The inequality holds for all bounded domains with $C^1$ boundary.

Equality in the inequality characterizes convex domains.

The inequality involves boundary integrals of normal vectors and boundary points.

Abstract

A property of smooth convex domains $\Omega \subset \mathbb{R}^n$ is that if two points on the boundary $x, y \in \partial \Omega$ are close to each other, then their normal vectors $n(x), n(y)$ point roughly in the same direction and this direction is almost orthogonal to $x-y$ (for `nearby' $x$ and $y$). We prove there exists a constant $c_n > 0$ such that if $\Omega \subset \mathbb{R}^n$ is a bounded domain with $C^1-$boundary $\partial \Omega$, then $$ \int_{\partial \Omega \times \partial \Omega} \frac{\left|\left\langle n(x), y - x \right\rangle \left\langle y - x, n(y) \right\rangle \right| }{\|x - y\|^{n+1}}~d \sigma(x) d\sigma(y) \geq c_n |\partial \Omega|$$ and equality occurs if and only if the domain $\Omega$ is convex.