Sharp L ¹-Poincaré inequalities correspond to optimal hypersurface cuts

Abstract

Let ${\Omega \subset \mathbb{R}^n}$ be a convex set. If ${u: \Omega \rightarrow \mathbb{R}}$ has mean 0, then we have the classical Poincaré inequality

$$\|u \|_{L^p} \leq c_p {\rm diam}(\Omega) \| \nabla u \|_{L^p}$$

with sharp constants ${c_2 = 1/\pi}$ (Payne and Weinberger, 1960) and c ₁ =  1/2 (Acosta and Duran, 2005) independent of the dimension. The sharp constants c _p for 1 <  p <  2 have recently been found by Valtorta (2012) and Ferone, Nitsch and 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 and Simonovits to show that

$$\left\|u\right\|_{L^{1}(\Omega)}\leq \frac{2}{\log{2}} M_{}(\Omega)\left\|\nabla u\right\|_{L^{1}(\Omega)},$$

where ${M(\Omega)}$ is the average distance between a point in ${\Omega}$ and the center of gravity of ${\Omega}$ . If ${\Omega}$ is a regular simplex in ${\mathbb{R}^n}$ , this yields an improvement by a factor of ${\sim \sqrt{n}}$ .