For convex domains of diameter d, Poincaré inequality holds with constant d/pi

You need to know: Euclidean space ${\mathbb R}^n$ , norm $||x||=\sqrt{(x,x)}$ of $x\in{\mathbb R}^n$ , convex domain $\Omega\subset {\mathbb R}^n$ , diameter of $\Omega$ , integral $\int_\Omega u(x) dx$ of function $u:\Omega \to {\mathbb R}$ , gradient $\nabla u: \Omega \to {\mathbb R}^n$ of u.

Background: For convex domain $\Omega\subset {\mathbb R}^n$ , let $H^1(\Omega)$ be the set of functions $u:\Omega \to {\mathbb R}$ for which $\nabla u$ exists on $\Omega$ and both $||u||_{L^2(\Omega)}=\sqrt{\int_\Omega u(x)^2 dx}$ and $||\nabla u||_{L^2(\Omega)}=\sqrt{\int_\Omega ||\nabla u(x)||^2 dx}$ are finite.

The Theorem: On 28th February 2003, Mario Bebendorf submitted to the Journal of Analysis and its Applications a paper in which he proved that, for any convex domain $\Omega\subset {\mathbb R}^n$ with diameter d, inequality $||u||_{L^2(\Omega)} \leq \frac{d}{\pi}||\nabla u||_{L^2(\Omega)}$ holds for all $u \in H^1(\Omega)$ such that $\int_\Omega u(x) dx = 0$ .

Short context: The inequality $||u||_{L^2(\Omega)} \leq c_\Omega||\nabla u||_{L^2(\Omega)}$ holds for any (non necessarily convex) subset $\Omega\subset {\mathbb R}^n$ , bounded at least in one direction. It is known as the Poincaré inequality. However, in this generality, it is unclear how to explicitly express constant $c_\Omega$ in terms of parameters of set $\Omega$ . In 1960, Payne and Weinberger published a theorem stating that if $\Omega\subset {\mathbb R}^n$ is convex with diameter d, then Poincaré inequality holds with $c_\Omega=\frac{d}{\pi}$ . However, their proof is correct only for $n=2$ . The Theorem proves this result for all n.