Inequalities for the Radon transform on convex sets

TL;DR

This paper explores inequalities related to the Radon transform on convex sets, extending classical problems like Busemann-Petty and slicing inequalities to more general measures and functions.

Contribution

It introduces a unifying inequality that encompasses previous results on the Radon transform in convex geometry, generalizing volume-based problems to arbitrary measures.

Findings

Extended Busemann-Petty problem to arbitrary measures

Established a slicing inequality for probability densities on convex bodies

Proved a unifying inequality for Radon transform properties

Abstract

Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex bodies including an extension of the Busemann-Petty problem and a slicing inequality for arbitrary functions. The latter means that the sup-norm of the Radon transform of any probability density on a convex body of volume one is bounded from below by a positive constant depending only on the dimension. In this note, we prove an inequality that serves as an umbrella for these results