An isomorphic version of the Busemann-Petty problem for arbitrary measures

TL;DR

This paper establishes an isomorphic version of the Busemann-Petty problem for arbitrary measures, providing bounds on measures of convex bodies based on hyperplane sections, with improved constants for specific cases.

Contribution

It extends the Busemann-Petty problem to arbitrary measures with new bounds and constants, including a hyperplane inequality for convex measures.

Findings

Proved an inequality relating measures of convex bodies and their hyperplane sections.

Derived improved constants for special classes of measures and bodies.

Established a hyperplane inequality for convex measures.

Abstract

We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le \sqrt{n} \mu(L).$ We also prove this result with better constants for some special classes of measures and bodies. Finally, we prove a version of the hyperplane inequality for convex measures.