On Lipschitz homogeneity of the Hilbert cube

Abstract:

The main contribution of this paper is to prove the conjecture of [Vä] 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, ${d_s}(x,y) = \sup \{ {s_k}|{x_k} - {y_k}|:k \in N\}$, 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.