Shape optimization under width constraint: the Cheeger constant and the torsional rigidity

TL;DR

This paper proves that among shapes with a fixed minimal width, the equilateral triangle optimizes the Cheeger constant and torsional rigidity, using comparison techniques and shape derivatives.

Contribution

It establishes the optimality of the equilateral triangle for these shape functionals under width constraints, a novel geometric optimization result.

Findings

Equilateral triangle maximizes the Cheeger constant.

Equilateral triangle minimizes torsional rigidity.

Comparison techniques with simpler shapes are effective for shape optimization.

Abstract

In this article it is shown that the equilateral triangle maximizes the Cheeger constant and minimizes the torsional rigidity among shapes having a fixed minimal width. The proof techniques use direct comparisons with simpler shapes, consisting of disks with three disjoint caps. Comparison results for harmonic functions help establish that in non-equilateral configurations the shape derivative has an appropriate sign, contradicting optimality.