On sharp bounds for marginal densities of product measures

TL;DR

This paper establishes sharp bounds on the marginal densities of product measures in high dimensions, showing that these bounds depend exponentially on the marginal's dimension, and introduces related isoperimetric inequalities.

Contribution

It provides the optimal constants for bounds on marginal densities of product measures and extends Ball's cube slicing approach to functions.

Findings

Bound on marginal density by 2^{k/2} for product measures

Extension of cube slicing method to functions

Isoperimetric inequality for averages of marginals

Abstract

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if $f$ is a probability density on $\R^n$ of the form $f(x)=\prod_{i=1}^n f_i(x_i)$, where each $f_i$ is a density on $\R$, say bounded by one, then the density of any marginal $\pi_E(f)$ is bounded by $2^{k/2}$, where $k$ is the dimension of $E$. The proof relies on an adaptation of Ball's approach to cube slicing, carried out for functions. Motivated by inequalities for dual affine quermassintegrals, we also prove an isoperimetric inequality for certain averages of the marginals of such $f$ for which the cube is the extremal case.