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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On Lipschitz homogeneity of the Hilbert cube


Author: Aarno Hohti
Journal: Trans. Amer. Math. Soc. 291 (1985), 75-86
MSC: Primary 54E40; Secondary 57N20
MathSciNet review: 797046
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Abstract: The main contribution of this paper is to prove the conjecture of [] that the Hilbert cube $ Q$ is Lipschitz homogeneous for any metric $ {d_s}$, where $ s$ is a decreasing sequence of positive real numbers $ {s_k}$ converging to zero, $ {d_s}(x,y) = \sup \{ {s_k}\vert{x_k} - {y_k}\vert:k \in N\} $, and $ R(s) = \sup \{ {s_k}/{s_{k + 1}}:k \in N\} < \infty $. In addition to other results, we shall show that for every Lipschitz homogeneous compact metric space $ X$ there is a constant $ \lambda < \infty $ such that $ X$ is homogeneous with respect to Lipschitz homeomorphisms whose Lipschitz constants do not exceed $ \lambda $. Finally, we prove that the hyperspace $ {2^I}$ of all nonempty closed subsets of the unit interval is not Lipschitz homogeneous with respect to the Hausdorff metric.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0797046-7
PII: S 0002-9947(1985)0797046-7
Keywords: Hilbert cube, homogeneous, Lipschitz homeomorphism, $ Q$-manifold, hyperspace
Article copyright: © Copyright 1985 American Mathematical Society