Bounding marginal densities via affine isoperimetry

Abstract

Let µ be a probability measure on 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 µ, any marginal π E (µ), or a small perturbation thereof, satisfies a nearly optimal small-ball probability.

Norm estimates for k-plane transforms and geometric inequalities

“…For applications of (4.18), the reader is referred to [10,13,15,26,27,50,51,70]. A similar inequality on the sphere and the hyperbolic spaces was studied by Dann, Kim and Yaskin [14].…”

Flag numbers and floating bodies

“…Many extension of both inequalities were since discovered. See for instance [23,27,37,77,71]. A lower bound for the volume product is a well-known open conjecture by Mahler [53,54].…”

Busemann's intersection inequality in hyperbolic and spherical spaces

“…In a slightly different form, (1) can be stated as follows. Centered ellipsoids in R n are the only maximizers of the quantity S n−1 |K ∩ ξ ⊥ | n dξ (2) in the class of star bodies of a fixed volume. In this paper we study this question in the hyperbolic space H n and the sphere S n (or, more precisely, a hemisphere S n + , as explained in the next section).…”

“…It is surprising that in S n + with n ≥ 3 centered balls are neither maximizers nor minimizers, even in the class of origin-symmetric convex bodies. We also obtain an 2 SUSANNA DANN, JAEGIL KIM, AND VLADYSLAV YASKIN optimal lower bound for (2) in the class of star bodies in S n + , n ≥ 3, of a given volume and describe the equality cases. Finally, we prove a version of Busemann's intersection inequality (together with the equality cases) for general measures on R n and H n .…”