On Poincaré Constants Related to Isoperimetric Problems in Convex Bodies

Abstract. For any convex set [Formula: see text], we provide a lower bound for the inverse of the Poincaré constant in [Formula: see text]: it refines an inequality in terms of the diameter due to Acosta and Durán, via the addition of an extra term taking account of the flatness of the domain. In dimension [Formula: see text], 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 an 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 center of the ball. As a key intermediate step, we prove that the ball maximizes the Poincaré constant in [Formula: see text], among convex bodies [Formula: see text] of given constant width.