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Some remarks about the maximal perimeter of convex sets with respect to probability measures
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 n with respect to various classes of measures. Firstly, we show that for a probability measure μ on n, 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.

更新日期:2020-08-09

 

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