Slicing inequalities for measures of convex bodies

TL;DR

This paper extends slicing inequalities to arbitrary measures and lower-dimensional sections of convex bodies, providing new bounds for specific classes and revealing differences from volume-based cases.

Contribution

It generalizes slicing inequalities to measures and lower-dimensional sections, proving results for unconditional bodies, duals with bounded volume ratio, and symmetric bodies under certain conditions.

Findings

Proved slicing inequalities for unconditional convex bodies.

Extended results to duals of bodies with bounded volume ratio.

Showed that minimal sections for some measures differ from volume cases.

Abstract

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove it for arbitrary symmetric convex bodies under the condition that the dimension of sections is less than $\lambda n$ for some $\lambda\in (0,1).$ The constant depends only on $\lambda.$ Finally, we show that the behavior of the minimal sections for some measures may be different from the case of volume.