A hyperplane inequality for measures of convex bodies in $\R^n,\ n\le 4$

TL;DR

This paper extends the hyperplane inequality to arbitrary measures for convex bodies in dimensions up to 4, and establishes stability results related to the Busemann-Petty problem.

Contribution

It generalizes the hyperplane inequality from volume to arbitrary measures in low dimensions and proves stability in the Busemann-Petty problem.

Findings

Hyperplane inequality holds for arbitrary measures in dimensions ≤ 4.

Stability results affirm that smaller hyperplane sections imply smaller measure.

Extension of Busemann-Petty problem solutions to broader measure settings.

Abstract

We generalize the hyperplane inequality in dimensions up to 4 to the setting of arbitrary measures in place of the volume. To prove this generalization we establish stability in the affirmative part of the solution to the Busemann-Petty problem asking whether symmetric convex bodies in R^n with smaller (n-1)-dimensional volume of all central hyperplane sections necessarily have smaller n-dimensional volume.