On the nonexistence of bilipschitz parameterizations and geometric problems about $A_{\infty }$-weights

Abstract

How can one recognize when a metric space is bilipschitz equivalent to a Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of ${\mathbb{R}}^n$ are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bilipschitz parameterizations are wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bilipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on ${\mathbb{R}}^n$ using an $A_{\infty }$ weight.