On the total surface area of potato packings

TL;DR

This work shows that in broad metric-measure spaces with doubling measures and a $(1,1)$-Poincaré inequality, a packing of open sets touching only on measure-zero sets must exhibit infinite total surface area. By formulating a perimeter-like functional $F$ with natural axioms and proving a key additivity/dichotomy theorem, the authors extend packing results from Euclidean spaces to PI-spaces, smooth Riemannian manifolds, and certain sub-Riemannian contexts. Consequently, packings force the residual set to have dimension at least $d-1$, providing sharp, generalizable insights into boundary behavior and geometric decomposition in diverse spaces.

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 Poincaré inequality, in particular in smooth Riemannian manifolds and even in some sub-Riemannian spaces.

Figures (1)

• Figure 1: On the left-hand side: a potato bag. On the right-hand side: an Apollonian gasket.

Theorems & Definitions (17)

• Lemma 1.1
• proof
• Definition 1.2
• Lemma 1.3
• proof
• Proposition 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1