An inequality characterizing convex domains

A property of smooth convex domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is that if two points on the boundary $x,y\in \mathrm{\partial }\mathrm{\Omega }$ 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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain with $C^1-$boundary $\mathrm{\partial }\mathrm{\Omega }$, then $${\int }_{\mathrm{\partial }\mathrm{\Omega }\times \mathrm{\partial }\mathrm{\Omega }}\frac{\mid ⟨n(x),y-x⟩⟨y-x,n(y)⟩\mid }{\mathrm{\parallel }x-y{\mathrm{\parallel }}^{n+1}}d\sigma (x)d\sigma (y)\ge c_n\mathrm{\mid }\mathrm{\partial }\mathrm{\Omega }\mathrm{\mid }$$ and equality occurs if and only if the domain $\mathrm{\Omega }$ is convex.