Uniformly Perfect Sets, Hausdorff Dimension, and Conformal Capacity

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 \(\mathbb {R}^n\) ${\mathbb{R}}^n$ 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 \({{\mathbb {R}}}^n\,\) ${\mathbb{R}}^n$ 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.