On a Cheeger--Kohler-Jobin inequality

TL;DR

This paper investigates a scale-invariant optimization problem involving torsional rigidity and Cheeger constant among convex sets, proving existence of solutions, conjecturing uniqueness, and exploring related inequalities and definitions.

Contribution

It establishes the existence of an optimal convex set for the problem, proposes a conjecture on the minimizer's uniqueness, and analyzes the equivalence of Cheeger constant definitions.

Findings

Existence of an optimal convex set for the minimization problem.

Conjecture that the ball is the unique minimizer.

A sufficient condition supporting the conjecture and a quantitative inequality for the Cheeger constant.

Abstract

We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely $\min T_2(\Omega) ^{\frac{1}{N+2}}h_1(\Omega)$ among open convex bounded sets $\Omega \subset \mathbb R^N$, where $T_2(\Omega)$ denotes the torsional rigidity of a set $\Omega$ and $h_1(\Omega)$ its Cheeger constant. We prove the existence of an optimal set and we conjecture that the ball is the unique minimizer. We provide a sufficient condition for the validity of the conjecture, and an application of the conjecture to prove a quantitative inequality for the Cheeger constant. We also show lack of existence for the problem above among several other classes of sets. As a side result we discuss the equivalence of the several definitions of Cheeger constants present in the literature and show a quite general class of sets for which those are equivalent.