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Abstract

We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. We observe that the compacta hyperspace $K(X)$ of any separable, uniformly disconnected metric space $X$ admits a bi-Lipschitz embedding in $\ell^2$. If $X$ is a countable compact metric space containing at most $n$ nonisolated points, there is a Lipschitz embedding of $K(X)$ in $\mathbb R^{n+1}$; in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. By way of contrast, the hyperspace $K([0,1])$ of the unit interval contains a bi-Lipschitz copy of a certain self-similar doubling series-parallel graph studied by Laakso, Lang–Plaut, and Lee–Mendel–Naor, and consequently admits no bi-Lipschitz embedding into any uniformly convex Banach space. Schori and West proved that $K([0,1])$ is homeomorphic with the Hilbert cube, while Hohti showed that $K([0,1])$ is not bi-Lipschitz equivalent with a variety of metric Hilbert cubes.