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 Rⁿ 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 Rⁿ\, 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.