The centered convex body whose marginals have the heaviest tails

TL;DR

This paper investigates the maximum ratio between different norms of marginals of centered convex bodies, identifying extremal cases like the simplex and exploring properties of these marginals in high-dimensional and log-concave settings.

Contribution

It characterizes the extremal marginals for the norm ratio problem, including the simplex, and introduces a family of distributions where the maximum is attained, with exact results for specific parameters.

Findings

The simplex maximizes the norm ratio among centered convex bodies.

A one-parameter family of distributions attains the maximum ratio in the log-concave case.

Exact maximizers are identified for p=2 and even q.

Abstract

Given any real numbers $1<p<q$, we study the norm ratio (i.e. the ratio between the $q$-norm and the $p$-norm) of marginals of centered convex bodies. We first show that some marginal of the simplex maximizes said ratio in the class of $n$-dimensional centered convex bodies. We then pass to the dimension independent (i.e. log-concave) case where we find a 1-parameter family of random variables in which the maximum ratio must be attained, and find the exact maximizer of the ratio when $p=2$ and $q$ is even. In addition, we find another interesting maximization property of marginals of the simplex involving functions with positive third derivatives.