"Gromov-Hausdorff Hyperspaces of $\mathbb R^n$"Antonyan, SergeyLet $(M, d)$ be a metric space. For two non-empty subsets $A$, $B\subset M$, the Hausdorff distance $d_H(A, B)$ is defined as follows: $ d_H(A, B)=\max\{\sup_{a\in A}d(a, B), \ \sup_{b\in B}d(b, A)\},$ where $d(x, C)=\inf\{d(x, c)\mid c\in C\}$. The set of all non-empty compact subsets of $M$ is denoted by $2^M$ and is endowed with the Hausdorff metric $d_H$. The pair $(2^M, d_H)$ is called the hyperspace of $M$. For two compact metric spaces $X$ and $Y$, their Gromov-Hausdorff distance $d_{GH} (X,Y )$ is defined to be the infimum of all real numbers $r>0$ such that here exist a metric space $(M, d)$ and isometric embeddings $ i:X\hookrightarrow M$ and $j :Y\hookrightarrow M$ with the Hausdorff distance $d_H(i(X ), j(Y ))$ less than $r$. It is a useful tool for studying topological properties of families of metric spaces. Clearly, the Gromov-Hausdorff distance between two isometric spaces is zero; it is a metric on the family $\rm{\mathbb {GH}}$ of isometry classes of compact metric spaces. The metric space $(\rm{\mathbb {GH}}, d_{GH})$ is called the Gromov-Hausdorff space. In this talk we mainly are interested in the subspace $\mathbb {GH}(\mathbb R^n)$ of $\mathbb {GH}$ consisting of the classes $[E]\in$$\mathbb {GH}$ whose representative $E$ is a metric subspace of the standard Euclidean space $\mathbb R^n, \ n\ge 1$. $\mathbb {GH}(\mathbb R^n)$ is called {the Gromov-Hausdorff hyperspace} of $\mathbb R^n$. One of the main results of this talk asserts that $\mathbb {GH}(\mathbb R^n)$ is homeomorphic to the orbit space $2^{\mathbb R^n}/E(n)$, where $2^{\mathbb R^n}$ is the hyperspace of all non-empty compact subsets of $\mathbb R^n$ endowed with the Hausdorff metric, and $E(n)$ is the isometry group of $\mathbb R^n$. This is applied to prove that $ \mathbb {GH}(\mathbb R^n)$ is homeomorphic to the punctured Hilbert cube $[0, 1]^{\aleph_0}\setminus\{*\}$. |
« back