On a convexity property of sections of the cross-polytope

TL;DR

This paper proves that the volume of central sections of dilated cross-polytopes exhibits log-concavity, extending the understanding of geometric inequalities related to convex bodies.

Contribution

It establishes the log-concavity of section volumes of dilated cross-polytopes, providing a new inequality for the Lebesgue measure on subspaces.

Findings

Proves log-concavity of section volumes of cross-polytopes.

Establishes the strong B-inequality for the cross-polytope.

Extends geometric inequalities to arbitrary subspace sections.

Abstract

We establish the log-concavity of the volume of central sections of dilations of the cross-polytope (the strong B-inequality for the cross-polytope and Lebesgue measure restricted to an arbitrary subspace).