Beyond Cheeger's constant

2024/04/05 by Lorenzo Brasco, Brasco, Lorenzo · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2404.03941

openalex publication_date 2024/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Cheeger constant of an open set of the Euclidean space is defined by minimizing the ratio "perimeter over volume", among all its smooth compactly contained subsets. We consider a natural variant of this problem, where the volume of admissible sets is raised to any positive power. We show that for \it sublinear powers, all these generalized Cheeger constants are equivalent to the standard one, by means of a universal two-sided estimate. We also show that this equivalence breaks down for \it superlinear powers. In this case, some weird phenomena appear. We finally consider the case of convex planar sets and prove an existence result, under optimal assumptions.

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