On Poincaré constants related to isoperimetric problems in convex bodies

Abstract

For any convex set $\Omega \subset {\mathbb{R}}^N$, we provide a lower bound for the inverse of the Poincaré constant in $W^{1,1}(\Omega )$: it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an extra term giving account for the flatness of the domain. In dimension $N=2$, we are able to make the extra term completely explicit, thus providing a new Bonnesen-type inequality for the Poincaré constant in terms of diameter and inradius. Such estimate is sharp, and it is asymptotically attained when the domain is the intersection of a ball with a strip bounded by parallel straight lines, symmetric about the centre of the ball. As a key intermediate step, we prove that the ball maximizes the Poincaré constant in $W^{1,1}(\Omega )$, among convex bodies $\Omega$ of given constant width.