Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A reverse isoperimetric inequality for the Cheeger constant under width constraint

Ilias Ftouhi, Ilaria Lucardesi, Giorgio Saracco

TL;DR

This work resolves a planar reverse isoperimetric-type question for the Cheeger constant under a minimal width constraint: among planar convex bodies, the product $w(K)h(K)$ is maximized by the equilateral triangle $T_e$, with equality only at $K=T_e$, which implies a sharp bound $h(K)\\le h(T_e)$ under fixed width. The authors provide two distinct proofs of this extremal inequality, extend the analysis to arbitrary dimensions proving existence of maximizers and ill-posedness of minimizers, and establish a quantitative stability: when the deficit $\\delta_{wh}(K)=w(T_e)h(T_e)-w(K)h(K)$ is small, the Hausdorff-width asymmetry $\\alpha_E(K)$ is controlled linearly by the deficit via a constant depending on a threshold $\\eta$. The results combine width/inner-parallel-set estimates, classical Cheeger-set characterizations (KLR05), and quantitative isoperimetric-type inequalities to yield a robust, sharp description of how near-extremal shapes must resemble an equilateral triangle in the Hausdorff sense.

Abstract

Henrot and Lucardesi, in Commun. Contemp. Math. (2024), conjectured that among planar convex sets with prescribed minimal width, the equilateral triangle uniquely maximizes the Cheeger constant. In this short note, we confirm this conjecture. Moreover, we establish a stability result for the inequality in terms of the Hausdorff distance.

A reverse isoperimetric inequality for the Cheeger constant under width constraint

TL;DR

This work resolves a planar reverse isoperimetric-type question for the Cheeger constant under a minimal width constraint: among planar convex bodies, the product is maximized by the equilateral triangle , with equality only at , which implies a sharp bound under fixed width. The authors provide two distinct proofs of this extremal inequality, extend the analysis to arbitrary dimensions proving existence of maximizers and ill-posedness of minimizers, and establish a quantitative stability: when the deficit is small, the Hausdorff-width asymmetry is controlled linearly by the deficit via a constant depending on a threshold . The results combine width/inner-parallel-set estimates, classical Cheeger-set characterizations (KLR05), and quantitative isoperimetric-type inequalities to yield a robust, sharp description of how near-extremal shapes must resemble an equilateral triangle in the Hausdorff sense.

Abstract

Henrot and Lucardesi, in Commun. Contemp. Math. (2024), conjectured that among planar convex sets with prescribed minimal width, the equilateral triangle uniquely maximizes the Cheeger constant. In this short note, we confirm this conjecture. Moreover, we establish a stability result for the inequality in terms of the Hausdorff distance.

Paper Structure

This paper contains 7 sections, 9 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Statement of main results
  4. Proofs of statements
  5. Maximization and minimization in arbitrary dimensions
  6. Proofs of Theorem 3.1
  7. Proof of Theorem 3.4

Key Result

Theorem 3.1

Let $K \in \mathcal{K}^2$ be a planar convex body. Then, Furthermore, equality holds if and only if $K$ is an equilateral triangle $T_e$.

Figures (1)

  • Figure 1: The polygons $K$ (in black) and $K_{-t}$ (in red), and $\alpha = \alpha_{\sigma(i)}/2$.

Theorems & Definitions (22)

  • Theorem 3.1: Reverse inequality & rigidity
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4: Local stability
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 12 more