Topological Hausdorff dimension and Poincaré inequality

Abstract

A relationship between Poincaré inequalities and the topological Hausdorff dimension is exposed—a lower bound on the dimension of Ahlfors regular spaces satisfying a weak (1, p)-Poincaré inequality is given.

Keywords

• Poincaré inequality
• metric space
• Cantor sets
• topological dimension
• Hausdorff dimension
• bi-Lipschitz map