![]() ![]() By way of contrast, the hyperspace K() 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. If X is a countable compact metric space containing at most n nonisolated points, there is a Lipschitz embedding of K(X) in ℝ n + 1 in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. We observe that the compacta hyperspace K(X) of any separable, uniformly disconnected metric space X admits a bi-Lipschitz embedding in ℓ². We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. We follow this by examining a geodesic bicombing on the nonempty compact subsets of X" role="presentation" style="box-sizing: inherit display: inline line-height: normal word-spacing: normal overflow-wrap: normal white-space: nowrap float: none direction: ltr max-width: none max-height: none min-width: 0px min-height: 0px border: 0px padding: 0px margin: 0px position: relative ">XX, assuming X" role="presentation" style="box-sizing: inherit display: inline line-height: normal word-spacing: normal overflow-wrap: normal white-space: nowrap float: none direction: ltr max-width: none max-height: none min-width: 0px min-height: 0px border: 0px padding: 0px margin: 0px position: relative ">XX is a proper metric space.Bi-Lipschitz embeddings of hyperspaces of compact sets ![]() ![]() If X" role="presentation" style="box-sizing: inherit display: inline line-height: normal word-spacing: normal overflow-wrap: normal white-space: nowrap float: none direction: ltr max-width: none max-height: none min-width: 0px min-height: 0px border: 0px padding: 0px margin: 0px position: relative ">XX is a normed space or an R" role="presentation" style="box-sizing: inherit display: inline line-height: normal word-spacing: normal overflow-wrap: normal white-space: nowrap float: none direction: ltr max-width: none max-height: none min-width: 0px min-height: 0px border: 0px padding: 0px margin: 0px position: relative ">RR-tree, this same method produces a consistent convex bicombing on CB(X)" role="presentation" style="box-sizing: inherit display: inline line-height: normal word-spacing: normal overflow-wrap: normal white-space: nowrap float: none direction: ltr max-width: none max-height: none min-width: 0px min-height: 0px border: 0px padding: 0px margin: 0px position: relative ">CB(X)CB(X). ![]() We show that if (X,d)" role="presentation" style="box-sizing: inherit display: inline line-height: normal word-spacing: normal overflow-wrap: normal white-space: nowrap float: none direction: ltr max-width: none max-height: none min-width: 0px min-height: 0px border: 0px padding: 0px margin: 0px position: relative ">(X,d)(X,d) is a metric space which admits a consistent convex geodesic bicombing, then we can construct a conical bicombing on CB(X)" role="presentation" style="box-sizing: inherit display: inline line-height: normal word-spacing: normal overflow-wrap: normal white-space: nowrap float: none direction: ltr max-width: none max-height: none min-width: 0px min-height: 0px border: 0px padding: 0px margin: 0px position: relative ">CB(X)CB(X), the hyperspace of nonempty, closed, bounded, and convex subsets of X" role="presentation" style="box-sizing: inherit display: inline line-height: normal word-spacing: normal overflow-wrap: normal white-space: nowrap float: none direction: ltr max-width: none max-height: none min-width: 0px min-height: 0px border: 0px padding: 0px margin: 0px position: relative ">XX (with the Hausdorff metric). ![]()
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