Sharp L^1 Poincare inequalities correspond to optimal hypersurface cuts

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a convex. If $u:\Omega \to \mathbb{R}$ has mean 0, then we have the classical Poincaré inequality $$ \|u \|_{L^p} \leq c_p \mbox{diam}(Ω) \| \nabla u \|_{L^p}$$ with sharp constants $c_2=1/\pi$ (Payne \& Weinberger, 1960) and $c_1=1/2$ (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, Lovász \& Simonovits to show that $${\Vert u\Vert }_{L^1(\Omega )}\le \frac{2}{\log 2}{M}_{}(\Omega ){\Vert \nabla u\Vert }_{L^1(\Omega )}$$ where $M_{}(\Omega )$ is the average distance between a point in $\Omega$ and the center of gravity of $\Omega$. If $\Omega$ 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 $\Omega \subset {\mathbb{R}}^n$ and every $u:\Omega \to \mathbb{R}$ with mean 0 $$ \left\|u\right\|_{L^{p}(Ω)}\leq \left(\frac{2}{\log{2}} M_{}(Ω) \right)^{\frac{1}{p}}\mbox{diam}(Ω)^{1-\frac{1}{p}}\left\|\nabla u\right\|_{L^{p}(Ω)}. $$

arXiv commentNew version with extension to L^p for p > 1, to be published in Archiv der Mathematik