Estimates for moments of general measures on convex bodies

TL;DR

This paper establishes new estimates for the moments of measures on convex bodies, leading to a slicing inequality, bounds on volume ratios, and a Busemann-Petty type result, advancing understanding in convex geometry.

Contribution

It introduces novel estimates for measure moments on convex bodies and applies them to derive inequalities and bounds related to convex geometric properties.

Findings

New slicing inequality for measures on convex bodies

Bounds on outer volume ratio distance to L_p-space unit balls

A Busemann-Petty type result for measure moments

Abstract

We prove several estimates for the moments of arbitrary measures on convex bodies. We apply these estimates to show a new slicing inequality for measures on convex bodies. We also deduce estimates for the outer volume ratio distance from an arbitrary centrally-symmetric convex body in R^n to the class of unit balls of n-dimensional subspaces of L_p-spaces. Finally, we prove a result of the Busemann-Petty type for these moments.