An inequality characterizing convex domains

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 $${\int }_{\partial \Omega \times \partial \Omega }\frac{\left|⟨n(x),y-x⟩⟨y-x,n(y)⟩\right|}{\Vert x-y{\Vert }^{n+1}}d\sigma (x)d\sigma (y)\ge c_n|\partial \Omega |$$ and equality occurs if and only if the domain $\Omega$ is convex.