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In 2010, Olson \& Robinson [Transactions of the American MathematicalSociety, 362(1), 145-168] introduced the notion of an almost homogeneous metricspace and showed that if $X$ is a subset of a Hilbert space such that $X-X$ isalmost homogeneous, then $X$ admits almost bi--Lipschitz embeddings intoEuclidean 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 theorigin, then $X$ can be embedded in a Euclidean space in an almostbi--Lipschitz way.

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