On isoperimetric local-Bollob\'as-Thomason inequalities

TL;DR

This paper establishes a new isoperimetric inequality relating convex bodies and Hanner polytopes, providing bounds on projection volumes that extend classical inequalities in convex geometry.

Contribution

It introduces a novel isoperimetric inequality connecting convex bodies with Hanner polytopes, expanding the understanding of projection volume bounds.

Findings

Proves an inequality linking convex bodies and Hanner polytopes.

Provides bounds on volumes of projections onto specific subspaces.

Extends classical isoperimetric inequalities in convex geometry.

Abstract

We prove the following isoperimetric-type inequality: for every convex body $K$ in $\mathbb R^n$ and some $\sigma\subset[n]:=\{1,\dots,n\}$ there exists a suitable Hanner polytope $B_K$ with the same volume as $K$ and such that the volume of each of its orthogonal projections onto every subspace whose basis is formed by the canonical vectors $\{e_i:i\in\tau\cup([n]\setminus\sigma)\}$, for every $\tau\subseteq\sigma$, bounds from below the volume of the corresponding projections of $K$.