Bi-Lipschitz embeddings of self-contracted curves into Euclidean spaces

Abstract: The question about characterization of metric spaces allowing bi-Lipschitz embeddings into Euclidean spaces is notoriously hard. To simplify the situation we deal with a 1-dimensional version of this question. Namely we prove that a doubling metric space can be embedded into Euclidean space under assumption that it can be presented as an image of a weakly self-contracting curve. (We say that a curve Г is weakly self-contracting if there exist K > 0 such that for every x < y < z < w in the domain of the curve we have |Г(y)Г(z)| < K|Г(x)Г(w)|, where |AB| denotes the distance between A and B.)

differential geometrymetric geometry

Audience: researchers in the topic

A.D. Alexandrov Geometry Seminar