Maximal surface area of polytopes with respect to log-concave rotation invariant measures

TL;DR

This paper investigates the maximal surface area of convex polytopes with a limited number of facets under rotation invariant log-concave measures, providing tight bounds and extending known results for specific measures.

Contribution

It derives tight bounds on the surface area of polytopes with K facets under log-concave measures, generalizing previous results for Gaussian measures.

Findings

Bound $oxed{rac{ ext{sqrt}(n)}{ ext{E}|X|} ext{sqrt}( ext{log} K)}$ for all $K$.

Existence of polytopes with surface area matching the bounds up to a $ ext{log} n$ factor.

Extension of bounds to measures $oxed{ ext{gamma}_p}$, generalizing Gaussian case.

Abstract

It was shown in \cite{GL} that the maximal surface area of a convex set in $\mathbb{R}^n$ with respect to a rotation invariant log-concave probability measure $\gamma$ is of order $\frac{\sqrt{n}}{\sqrt[4]{Var|X|} \sqrt{\mathbb{E}|X|}}$, where $X$ is a random vector in $\mathbb{R}^n$ distributed with respect to $\gamma$. In the present paper we discuss surface area of convex polytopes $P_K$ with $K$ facets. We find tight bounds on the maximal surface area of $P_K$ in terms of $K$. We show that $\gamma(\partial P_K)\lesssim \frac{\sqrt{n}}{\mathbb{E}|X|}\cdot\sqrt{\log K}\cdot\log n$ for all $K$. This bound is better then the general bound for all $K\in [2,e^{\frac{c}{{\sqrt{Var|X|}}}}]$. Moreover, for all $K$ in that range the bound is exact up to a factor of $\log n$: for each $K\in [2,e^{\frac{c}{{\sqrt{Var|X|}}}}]$ there exists a polytope $P_K$ with at most $K$ facets such that $\gamma(\partial P_K)\gtrsim \frac{\sqrt{n}}{\mathbb{E}|X|}\sqrt{\log K}.$ %For the measures $\gamma_p$ with densities $C_{n,p} e^{-\frac{|y|^p}{p}}$ (where $p>0$) we obtain: $\gamma_p(\partial P_K)\lesssim \frac{\sqrt{n}}{\mathbb{E}X}\sqrt{\log K},$ which was obtained for the standard Gaussian measure $\gamma_2$ by F. Nazarov.