  We introduce the concept of hereditarily non uniformly perfect sets, compact
sets for which no compact subset is uniformly perfect, 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, we
give an example of a compact set in the plane of Hausdorff dimension 2 (and
positive logarithmic capacity) which is hereditarily non uniformly perfect.
