Uniformly perfect sets, Hausdorff dimension, and conformal capacity

Abstract

Using the definition of uniformly perfect sets in terms of convergent sequences, we apply lower bounds for the Hausdorff content of a uniformly perfect subset $E$ of $\mathbb{R}^n$ to prove new explicit lower bounds for the Hausdorff dimension of $E.$ These results also yield lower bounds for capacity test functions, which we introduce, and enable us to characterize domains of $\mathbb{R}^n$ with uniformly perfect boundaries. Moreover, we show that an alternative method to define capacity test functions can be based on the Whitney decomposition of the domain considered.