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.  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10971125).
Author NameAffiliation
Zhilang ZHANG Department of Mathematics, Shantou University, Guangdong 515063, P. R. China 
Zhongqiang YANG Department of Mathematics, Shantou University, Guangdong 515063, P. R. China 
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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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