Accepted Paper
Inserted: 11 nov 2024
Last Updated: 4 mar 2025
Journal: Proc. Amer. Math. Soc.
Year: 2024
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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