Sharp L^1 Poincare inequalities correspond to optimal hypersurface cuts

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a convex. If $u: \Omega \rightarrow $\mathbb{R}$$hasmean0,thenwehavetheclassicalPoincar\acute{e}inequality$$\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p}\le c_{p}diam(\mathrm{\Omega })\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p}$$ with sharp constants $c_2=1\mathrm{/}\pi$ (Payne & Weinberger, 1960) and $c_1=1\mathrm{/}2$ (Acosta & Duran, 2005) independent of the dimension. The sharp constants $c_p$ for $1 < p < 2$ have recently been found by Ferone, Nitsch & Trombetti (2012). The purpose of this short paper is to prove a much stronger inequality in the endpoint $L^1$: we combine results of Cianchi and Kannan, Lov'{a}sz & Simonovits to show that $${\parallel u\parallel }_{L^1(\mathrm{\Omega })}\le \frac{2}{\log 2}{M}_{}(\mathrm{\Omega }){\parallel \mathrm{\nabla }u\parallel }_{L^1(\mathrm{\Omega })}$$ where $M_{}(\mathrm{\Omega })$ is the average distance between a point in $\mathrm{\Omega }$ and the center of gravity of $\mathrm{\Omega }$. If $\mathrm{\Omega }$ is a simplex, this yields an improvement by a factor of $\sim \sqrt{n}$ in $n$ dimensions. By interpolation, this implies that that for every convex $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and every $u:\mathrm{\Omega }\to \mathbb{R}$ with mean 0 $${\parallel u\parallel }_{L^p(\mathrm{\Omega })}\le {(\frac{2}{\log 2}{M}_{}(\mathrm{\Omega }))}^{\frac{1}{p}}diam(\mathrm{\Omega }{)}^{1-\frac{1}{p}}{\parallel \mathrm{\nabla }u\parallel }_{L^p(\mathrm{\Omega })}\mathrm{.}$$