On sharp bounds for marginal densities of product measures

Understand

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.