Isoperimetric and Poincar\'e inequalities on non-self-similar Sierpi\'nski sponges: the borderline case

TL;DR

This paper constructs new examples of subsets in Euclidean space that support a 1-Poincaré inequality with empty interior, using innovative methods involving relative isoperimetric inequalities and extending previous results to higher dimensions.

Contribution

It introduces a novel approach to establishing Poincaré inequalities via relative isoperimetric inequalities and generalizes existing constructions to broader classes of shapes and higher dimensions.

Findings

Supports 1-Poincaré inequality with empty interior

Extends non-self-similar Sierpiński sponge results to higher dimensions

Introduces new tools using isoperimetric inequalities at density levels

Abstract

In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincar\'e inequality yet have empty interior. These examples are formed from an iterative process that involves removing well-behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes families of curves, we include a new way of obtaining Poincar\'e inequalities through the use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we further introduce the notion of of isoperimetric inequalities at given density levels and a way to iterate such inequalities. These tools are presented and apply to general metric measure measures. Our examples subsume the previous results of Mackay, Tyson, and Wildrick regarding non-self similar Sierpi\'nski carpets, and extend them to many more general shapes as well as higher dimensions.