Extensions of the Busemann-Petty Problem for Arbitrary Measures

TL;DR

This paper extends the Busemann-Petty problem to arbitrary measures with diverse distributions, providing new conditions for when smaller hyperplane sections imply smaller total measure, and illustrating cases beyond previous results.

Contribution

It broadens the class of distributions for which the Busemann-Petty problem can be answered affirmatively, allowing different densities for sections and volumes.

Findings

Extended the problem to broader distribution classes

Allowed different densities for sections and volumes

Provided examples outside previous coverage

Abstract

The classical Busemann-Petty problem asks whether smaller central hyperplane sections of origin-symmetric convex bodies necessarily imply smaller total volume. Zvavitch studied this question for arbitrary measures with continuous even densities, providing sufficient conditions for affirmative cases in terms of the distributional behavior of the ratio between the densities involved. We refine this result by extending it to a broader class of distributions and allowing a distinct pair of densities for each body, one for hyperplane sections and another for the full volume. We also present some examples illustrating cases not covered by previous results.