On isoperimetric local-Bollob\'as-Thomason inequalities

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 $K$.