Sharp L^1 Poincare inequalities correspond to optimal hypersurface cuts

Let $Ω\subset \mathbb{R}^n$ be a convex. If $u: Ω\rightarrow \mathbb{R}$ 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 $c_2 = 1/π$ (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 $\left\|u\right\|_{L^{1}(Ω)} \leq \frac{2}{\log{2}} M_{}(Ω) \left\|\nabla u\right\|_{L^{1}(Ω)}$ where $M_{}(Ω)$ is the average distance between a point in $Ω$ and the center of gravity of $Ω$. If $Ω$ 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 $Ω\subset \mathbb{R}^n$ and every $u:Ω\rightarrow \mathbb{R}$ with mean 0 $$ \left\|u\right\|_{L^{p}(Ω)}\leq \left(\frac{2}{\log{2}} M_{}(Ω) \right)^{\frac{1}{p}}\mbox{diam}(Ω)^{1-\frac{1}{p}}\left\|\nabla u\right\|_{L^{p}(Ω)}. $$

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