On the Gaussian behavior of marginals and the mean width of random polytopes

TL;DR

This paper investigates the Gaussian-like behavior of marginals and the mean width of random polytopes within isotropic convex bodies, providing new bounds and extending known results in high-dimensional convex geometry.

Contribution

It establishes the order of the expected mean width of random polytopes and extends the understanding of 1-dimensional marginals in isotropic convex bodies.

Findings

Expected mean width is of order √(log N) L_K for N in specified range.

Extended the interval where marginals exhibit sub- or supergaussian behavior.

Provided new bounds related to the Gaussian behavior of marginals.

Abstract

We show that the expected value of the mean width of a random polytope generated by $N$ random vectors ($n\leq N\leq e^{\sqrt n}$) uniformly distributed in an isotropic convex body in $\R^n$ is of the order $\sqrt{\log N} L_K$. This completes a result of Dafnis, Giannopoulos and Tsolomitis. We also prove some results in connection with the 1-dimensional marginals of the uniform probability measure on an isotropic convex body, extending the interval in which the average of the distribution functions of those marginals behaves in a sub- or supergaussian way.