  In this paper we construct a large family of examples of subsets of Euclidean
space that support a 1-Poincar\'e inequality yet have empty interior. These
examples are formed from an iterative process that involves removing
well-behaved domains, or more precisely, domains whose complements are uniform
in the sense of Martio and Sarvas.
  While existing arguments rely on explicit constructions of Semmes families of
curves, we include a new way of obtaining Poincar\'e inequalities through the
use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we
further introduce the notion of of isoperimetric inequalities at given density
levels and a way to iterate such inequalities. These tools are presented and
apply to general metric measure measures.
  Our examples subsume the previous results of Mackay, Tyson, and Wildrick
regarding non-self similar Sierpi\'nski carpets, and extend them to many more
general shapes as well as higher dimensions.
