In this note we study the maximal perimeter of a convex set in $\mathbb{ R}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $\mu$ on $ \mathbb{ R}^n$, satisfying very mild assumptions, there exists a convex set of $\mu$-perimeter at least $C\frac{ \sqrt{ n}}{ \sqrt[4]{ Var|X|} \sqrt{ \mathbb{ E}|X|}}.$ This implies, in particular, that for any isotropic log-concave measure $\mu$ one may find a convex set of $\mu$- perimeter of order $n^{ \frac{ 1}{ 8}}$. Secondly, we derive a general upper bound of $Cn|| f||^{ \frac{ 1}{ n}}_{ \infty}$ on the maximal perimeter of a convex set with respect to any log-concave measure with density $f$ 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 $n^2$ for the maximal $\mu$-perimeter of a convex set.
Categories: math.MG math.CA math.PR