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 $ℝn$ with respect to various classes of measures. Firstly, we show that for a probability measure $μ$ on $ℝn$, satisfying very mild assumptions, there exists a convex set of $μ$-perimeter at least $CnVar|X|4𝔼|X|.$ This implies, in particular, that for any isotropic log-concave measure $μ$, one may find a convex set of $μ$-perimeter of order $n18$. Secondly, we derive a general upper bound of $Cn||f||∞1n$ 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 $n2$ for the maximal $μ$-perimeter of a convex set.

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