On Non-Bi-Lipschitz Homogeneity of Some Hyperspaces

DOI：10.3770/j.issn:2095-2651.2014.03.014

 作者 单位 张志朗 汕头大学数学系, 广东 汕头 515063 杨忠强 汕头大学数学系, 广东 汕头 515063

设$(X,d)$是度量空间,如果对任意的$x, y\in X$,存在$X$到自身的同胚$h$使得$h(x)=y$且$h$和它的逆都是Lipschitz的,则称$(X,d)$为双Lipschitz齐次的.用$2^{(X,d)}$表示带有Hausdorf度量的$(X,d)$的非空紧子集全体所构成的度量空间.1985年Hohti证明了$2^{([0,1],d)}$不是双Lipschitz齐次的,这里$d$为$[0,1]$上的标准度量.本文从两个方向推广了这个结果.一个是对于满足一定条件的$[0,1]$上的相容度量$\rho,2^{([0,1],\rho)}$也不是双Lipschitz齐次的;另一个是如果度量空间$(X,d)$中存在一个非空开子空间等距同胚于欧氏空间$\mathbb{R}^m$的开子空间,则$2^{(X,d)}$不是双 Lipschitz齐次的.

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 $\mathbb{R}^m$.