On the nonexistence of bilipschitz parameterizations and geometric problems about \(A_ \infty\)-weights
Summary: How can one recognize when a metric space is bi-Lipschitz equivalent to a Euclidean space? One should not take that 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 bi-Lipschitz 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 bi-Lipschitz parameterizations are wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bi-Lipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on \(\mathbb{R}^n\) using an \(A_\infty\) weight.
bi-Lipschitz equivalent to a Euclidean space
scientific article; zbMATH DE number 933288