Hereditarily non uniformly perfect sets (Q2305704)

A compact set \(E\subset \mathbb{R}^n\) is said to be hereditarily non-uniformly perfect if no compact subset of it is uniformly perfect. The authors investigate various properties of hereditarily non-uniformly perfect sets and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, a planar compact set is explicitely constructed which has Hausdorff dimension 2 and is hereditarily non uniformly perfect.