2 results
Search Results
Now showing 1 - 2 of 2
Article Dvoretzky-Type Theorem for Locally Finite Subsets of a Hilbert Space(Annales Inst Fourier, 2025) Catrina, Florin; Ostrovska, Sofiya; Ostrovskii, Mikhail I.The main result of the paper: Given any epsilon > 0, every locally finite subset of l(2) admits a (1 + epsilon)-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which are of independent interest: (1) A direct sum of two finite-dimensional Euclidean spaces contains a sub-sum of a controlled dimension which is epsilon-close to a direct sum with respect to a 1-unconditional basis in a two-dimensional space. (2) For any finite-dimensional Banach space Y and its direct sum X with itself with respect to a 1-unconditional basis in a two-dimensional space, there exists a (1 + epsilon)-bilipschitz embedding of Y into X which on a small ball coincides with the identity map onto the first summand and on the complement of a large ball coincides with the identity map onto the second summand.Article Citation - WoS: 4Citation - Scopus: 4Nonexistence of Embeddings With Uniformly Bounded Distortions of Laakso Graphs Into Diamond Graphs(Elsevier Science Bv, 2017) Ostrovska, Sofiya; Ostrovskii, Mikhail I.Diamond graphs and Laakso graphs are important examples in the theory of metric embeddings. Many results for these families of graphs are similar to each other. In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. The well-known fact that Laakso graphs are uniformly doubling but diamond graphs are not, immediately implies that diamond graphs do not admit uniformly bilipschitz embeddings into Laakso graphs. The main goal of this paper is to prove that Laakso graphs do not admit uniformly bilipschitz embeddings into diamond graphs. (C) 2016 Elsevier B.V. All rights reserved.
