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On Lipschitz Homogeneity of the Hilbert Cube

Aarno Hohti
Transactions of the American Mathematical Society
Vol. 291, No. 1 (Sep., 1985), pp. 75-86
DOI: 10.2307/1999895
Stable URL: http://www.jstor.org/stable/1999895
Page Count: 12
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On Lipschitz Homogeneity of the Hilbert Cube
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Abstract

The main contribution of this paper is to prove the conjecture of $\lbrack V \ddot a\rbrack$ that the Hilbert cube Q is Lipschitz homogeneous for any metric ds, where s is a decreasing sequence of positive real numbers sk converging to zero, $d_s(x,y) = \sup\{s_k|x_k - yk|: k \in N\}$, 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 λ. Finally, we prove that the hyperspace 2I of all nonempty closed subsets of the unit interval is not Lipschitz homogeneous with respect to the Hausdorff metric.

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