Preprint
Inserted: 5 apr 2024
Last Updated: 8 apr 2024
Pages: 33
Year: 2024
Notes:
In memory of Rodolfo Boschi
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.
Keywords: convex sets, Sobolev-Poincaré Inequalities, Inradius, Cheeger's constant
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