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

## TL;DR

This paper investigates the maximal perimeter of convex sets under various probability measures, establishing bounds that depend on the measure's properties and providing insights into geometric measure theory.

## Contribution

It introduces new bounds for the maximal perimeter of convex sets with respect to different classes of measures, including log-concave and isotropic measures, extending previous geometric results.

## Key findings

- Existence of convex sets with perimeter at least proportional to ^{1/8} for isotropic log-concave measures.
- Upper bounds on maximal perimeter depending on measure density and dimension.
- Bounds are tight for certain distributions like the normal and uniform measures.

## Abstract

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.