Some remarks about the maximal perimeter of convex sets with respect to probability measures

Abstract

In this note, we study the maximal perimeter of a convex set in [Formula: see text] with respect to various classes of measures. Firstly, we show that for a probability measure [Formula: see text] on [Formula: see text], satisfying very mild assumptions, there exists a convex set of [Formula: see text]-perimeter at least [Formula: see text] This implies, in particular, that for any isotropic log-concave measure [Formula: see text], one may find a convex set of [Formula: see text]-perimeter of order [Formula: see text]. Secondly, we derive a general upper bound of [Formula: see text] on the maximal perimeter of a convex set with respect to any log-concave measure with density [Formula: see text] in an appropriate position. Our lower bound is attained for a class of distributions including the standard normal distribution. Our upper bound is attained, say, for a uniform measure on the cube. In addition, for isotropic log-concave measures, we prove an upper bound of order [Formula: see text] for the maximal [Formula: see text]-perimeter of a convex set.