Almost bi-Lipschitz embeddings using covers of balls centred at the origin

In Olson and Robinson (2010) N introduced the notion of an almost homogeneous metric space and showed that if X is a subset of a Hilbert space such that X - X is almost homogeneous, then X admits almost bi-Lipschitz embeddings into Euclidean spaces. In this paper, we extend this result and we show that if X is a subset of a Banach space such that X - X is almost homogeneous at the origin, then X can be embedded in a Euclidean space in an almost bi-Lipschitz way. (C) 2020 Elsevier Inc. All rights reserved.