On isoperimetric local-Bollobás-Thomason inequalities

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 $K$.