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

2025/06/09 by Beniamin Bogosel, Bogosel, Beniamin
Engineering · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC) #Structural Analysis and Optimization #Topology Optimization in Engineering

paper · pdf · doi:10.48550/arxiv.2506.07708

openalex publication_date 2025/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• A reverse isoperimetric inequality for the Cheeger constant under width constraint
• Three quantitative versions of the Pál inequality
• Numerical shape optimization among convex sets
• A Blaschke-Lebesgue Theorem for the Cheeger constant
• On the relations between principal eigenvalue and torsional rigidity
• Parametric Shape Optimization using the Support Function
• On two functionals involving the maximum of the torsion function
• Partial Differential Equations An Introduction
• Polygons as optimal shapes with convexity constraint

Related

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• Parametric Shape Optimization using the Support Function
• A shape optimization problem for the p-Laplacian
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• A Linear View on Shape Optimization