On Poincaré Constants Related to Isoperimetric Problems in Convex Bodies

For any convex set Ω⊂ R N, we provide a lower bound for the inverse of the Poincaré constant in W ^ 1, 1(Ω): 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 (Ω), among convex bodies $Ω$ of given constant width.