Shape optimization under width constraint: the Cheeger constant and the torsional rigidity
Optimization and Control
2026-03-24 v2 Metric Geometry
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.
Cite
@article{arxiv.2506.07708,
title = {Shape optimization under width constraint: the Cheeger constant and the torsional rigidity},
author = {Beniamin Bogosel},
journal= {arXiv preprint arXiv:2506.07708},
year = {2026}
}