On the total surface area of potato packings

TL;DR

This paper proves that filling a bag with infinitely many potatoes touching each other results in an infinite total surface area, extending the result to various geometric spaces.

Contribution

It establishes a general mathematical result about the total surface area of densely packed measurable sets in Euclidean and more general metric measure spaces.

Findings

Total surface area must be infinite for infinite dense packings

Result holds in Euclidean spaces, Riemannian manifolds, and some sub-Riemannian spaces

Applicable to measurable subsets with minimal contact points

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 the general context of doubling (even locally) metric measure spaces satisfying Poincar\'e inequality, in particular in smooth Riemannian manifolds and even in some sub-Riemannian spaces.