Semmes, Stephen On the nonexistence of bilipschitz parameterizations and geometric problems about \(A_ \infty\)-weights. (English) Zbl 0858.46017 Rev. Mat. Iberoam. 12, No. 2, 337-410 (1996). 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. Cited in 3 ReviewsCited in 86 Documents MSC: 46B20 Geometry and structure of normed linear spaces 46B07 Local theory of Banach spaces Keywords:bi-Lipschitz equivalent to a Euclidean space; bi-Lipschitz parameterizations; deforming the Euclidean metric; \(A_ \infty\) weight PDFBibTeX XMLCite \textit{S. Semmes}, Rev. Mat. Iberoam. 12, No. 2, 337--410 (1996; Zbl 0858.46017) Full Text: DOI EuDML