On Non-Bi-Lipschitz Homogeneity of Some Hyperspaces |
Received:March 21, 2013 Revised:January 14, 2014 |
Key Words:
non-bi-Lipschitz homogeneity hyperspace Hilbert cube.
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Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10971125). |
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Abstract: |
A metric space $(X, d)$ is called bi-Lipschitz homogeneous if for any points $x,y\in X$, there exists a self-homeomorphism $h$ of $X$ such that both $h$ and $h^{-1}$ are Lipschitz and $h(x)=y$. Let $2^{(X,d)}$ denote the family of all non-empty compact subsets of metric space $(X,d)$ with the Hausdorff metric. In 1985, Hohti proved that $2^{([0,1],d)}$ is not bi-Lipschitz homogeneous, where $d$ is the standard metric on $[0,1]$. We extend this result in two aspects. One is that $2^{([0,1],\varrho)}$ is not bi-Lipschitz homogeneous for an admissible metric $\varrho$ satisfying some conditions. Another is that $2^{(X,d)}$ is not bi-Lipschitz homogeneous if $(X,d)$ has a nonempty open subspace which is isometric to an open subspace of $m$-dimensional Euclidean space $\mathbb{R}^m$. |
Citation: |
DOI:10.3770/j.issn:2095-2651.2014.03.014 |
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