Measure comparison and distance inequalities for convex bodies

TL;DR

This paper establishes new measure comparison inequalities for convex bodies, providing insights into slicing and distance inequalities, and offers a sharp estimate for the outer volume ratio distance to $L_p$ unit balls.

Contribution

It introduces novel versions of the isomorphic Busemann-Petty problem for different measures and applies these to derive slicing and distance inequalities, along with a sharp outer volume ratio estimate.

Findings

New measure comparison inequalities for convex bodies.

Recovery of slicing and distance inequalities.

Sharp upper estimate for outer volume ratio distance.

Abstract

We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio distance from an arbitrary convex body to the unit balls of subspaces of $L_p$.