Bounding marginal densities via affine isoperimetry

TL;DR

This paper establishes bounds on the marginals of probability densities in high dimensions using affine isoperimetric inequalities, revealing a trade-off for product measures and providing nearly optimal small-ball probability estimates.

Contribution

It introduces new affinely-invariant extremal inequalities for averages of functions on Grassmannians, extending dual affine quermassintegrals to a functional setting.

Findings

Most marginals of bounded densities are well-bounded on high-dimensional subspaces.

For product measures, bounds depend on the probability with which they hold.

Any marginal or small perturbation thereof satisfies a nearly optimal small-ball probability.

Abstract

Let $\mu$ be a probability measure on $\mathbb{R}^n$ with a bounded density $f$. We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely-invariant extremal inequalities for certain averages of $f$ on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann--Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on $\mu$, any marginal $\pi_E(\mu)$, or a small perturbation thereof, satisfies a nearly optimal small-ball probability.