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On isoperimetric local-Bollobás-Thomason inequalities

Luis J. Alías, Bernardo González Merino, Beatriz Marín Gimeno

TL;DR

The paper addresses a local isoperimetric-type problem for convex bodies by comparing projections of a given $K\in\mathcal{K}^n$ with those of a suitably chosen Hanner polytope $B_K$ of the same volume. The main approach constructs $B_K$ so that $\mathrm{vol}(P_{H_\sigma^{\bot}\oplus H_\tau}K)\ge \mathrm{vol}(P_{H_\sigma^{\bot}\oplus H_\tau}B_K)$ for all $\tau\subseteq\sigma$, and proves this via local Bollobás–Thomason inequalities, irreducible/induced covers, and a sequence of Brunn–Minkowski and Berwald-type arguments. A complete equality characterization is provided, describing when the projection inequalities are tight through induced covers and hypograph equalities, and the Appendix contains a self-contained proof of the central inequality. These results advance the understanding of projection-based isoperimetric phenomena and connect to extremizers linked to Hanner polytopes and Mahler-type questions in convex geometry.

Abstract

We prove the following isoperimetric-type inequality: for every convex body $K$ in $\mathbb R^n$ and some $σ\subset[n]:=\{1,\dots,n\}$ 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 $\{e_i:i\inτ\cup([n]\setminusσ)\}$, for every $τ\subseteqσ$, bounds from below the volume of the corresponding projections of $K$.

On isoperimetric local-Bollobás-Thomason inequalities

TL;DR

The paper addresses a local isoperimetric-type problem for convex bodies by comparing projections of a given with those of a suitably chosen Hanner polytope of the same volume. The main approach constructs so that for all , and proves this via local Bollobás–Thomason inequalities, irreducible/induced covers, and a sequence of Brunn–Minkowski and Berwald-type arguments. A complete equality characterization is provided, describing when the projection inequalities are tight through induced covers and hypograph equalities, and the Appendix contains a self-contained proof of the central inequality. These results advance the understanding of projection-based isoperimetric phenomena and connect to extremizers linked to Hanner polytopes and Mahler-type questions in convex geometry.

Abstract

We prove the following isoperimetric-type inequality: for every convex body in and some there exists a suitable Hanner polytope with the same volume as and such that the volume of each of its orthogonal projections onto every subspace whose basis is formed by the canonical vectors , for every , bounds from below the volume of the corresponding projections of .
Paper Structure (4 sections, 6 theorems, 72 equations)

This paper contains 4 sections, 6 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Isoperimetric local Bollobás-Thomason inequalities
  3. Characterization of equality case of local Bollobás-Thomason inequalities
  4. Appendix

Key Result

Theorem 1.1

Let $K\in\mathcal{K}^n$ and $\sigma\subset [n]$. Then, there exist some $c_i>0$, $i\in\sigma$, and a convex body $B_K$ of the form such that where $L$ is any $(n-|\sigma|)$-dimensional convex body contained in $H_\sigma^\bot.$

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • proof : Proof of Theorem \ref{['thm: isoperimetric main projections']}
  • Corollary 2.1
  • Lemma 2.2
  • proof
  • Remark
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:LocalBollobasThomasonineq']}
  • ...and 2 more