Isoperimetric and Poincaré Inequalities on Non-Self-Similar Sierpiński Sponges: the Borderline Case

Abstract

In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré 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.

Keywords

• isoperimetry
• sierpiński carpet
• self-improvement
• poincaré
• 26a45
• 30l99 (28a75; 28a80; 31e05)