On a Loomis-Whitney Type Inequality for Permutationally Invariant Unconditional Convex Bodies

TL;DR

This paper proves a log-concavity property of volume sequences derived from permutationally invariant unconditional convex bodies, extending geometric inequalities in high-dimensional convex analysis.

Contribution

It establishes a new Loomis-Whitney type inequality for a class of symmetric convex bodies, revealing novel volume sequence properties.

Findings

Volume sequence (|K_1|, ..., |K_n|) is log-concave.

Extends classical inequalities to permutationally invariant bodies.

Provides new insights into convex body projections and volume relations.

Abstract

For a permutationally invariant unconditional convex body K in R^n we define a finite sequence (K_j), j = 1, ..., n of projections of the body K to the space spanned by first j vectors of the standard basis of R^n. We prove that the sequence of volumes (|K_1|, ..., |K_n|) is log-concave.