Communications in Contemporary Mathematics ( IF 1.278 ) Pub Date : 2020-07-27 , DOI: 10.1142/s0219199720500376 Galyna V. Livshyts
In this note, we study the maximal perimeter of a convex set in with respect to various classes of measures. Firstly, we show that for a probability measure on , satisfying very mild assumptions, there exists a convex set of -perimeter at least This implies, in particular, that for any isotropic log-concave measure , one may find a convex set of -perimeter of order . Secondly, we derive a general upper bound of on the maximal perimeter of a convex set with respect to any log-concave measure with density 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 for the maximal -perimeter of a convex set.