Published Paper

Inserted: 11 nov 2024
Last Updated: 22 aug 2025

Journal: Proc. Amer. Math. Soc.
Volume: 153
Number: 10
Pages: 4327-4335
Year: 2025
Doi: 10.1090/proc/17277

Abstract:

We prove that if we fill without gaps a bag with infinitely many potatoes, in such a way that they touch each other in few points, then the total surface area of the potatoes must be infinite. In this context potatoes are measurable subsets of the Euclidean space, the bag is any open set of the same space. As we show, this result also holds in a fairly general context of doubling (even locally) metric measure spaces satisfying Poincaré inequality, in particular in smooth Riemannian manifolds and even in some sub-Riemannian spaces.

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