Abstract We investigate the hyperspace GH ( R n ) of the isometry classes of all non-empty compact subsets of a Euclidean space in the Gromov-Hausdorff metric. It is proved that for any n ≥ 1 , GH ( R n ) is homeomorphic to the orbit space 2 R n / E ( n ) of the hyperspace 2 R n of all non-empty compact subsets of a Euclidean space R n equipped with the Hausdorff metric and the natural action of the Euclidean group E ( n ) . This is further applied to prove that 2 R n / E ( n ) is homeomorphic to the open cone O C o n e ( C h ( B n ) / O ( n ) ) , where C h ( B n ) stands for the set of all A ∈ 2 R n for which the closed Euclidean unit ball B n is the least circumscribed ball (the Chebyshev ball). These results lead to determine the complete topological structure of GH ( R n ) for n ≤ 2 , namely, we prove that GH ( R n ) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n ≤ 2 , GH ( B n ) is homeomorphic to the Hilbert cube.

中文翻译：

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欧几里得空间的 Gromov-Hausdorff 超空间

摘要 我们研究了 Gromov-Hausdorff 度量中欧几里得空间的所有非空紧致子集的等距类的超空间 GH ( R n )。证明对于任何 n ≥ 1 , GH ( R n ) 同胚于欧氏空间 R n 的所有非空紧致子集的超空间 2 R n 的轨道空间 2 R n / E ( n ) Hausdorff 度量和欧几里得群 E(n) 的自然作用。这进一步用于证明2 R n / E ( n ) 同胚于开锥OC one ( C h ( B n ) / O ( n ) ) ，其中C h ( B n ) 代表所有的集合。 A ∈ 2 R n 其中闭欧几里得单位球 B n 是最小外接球（切比雪夫球）。这些结果导致确定 GH ( R n ) 对于 n ≤ 2 的完整拓扑结构，即，我们证明 GH ( R n ) 同胚于 Hilbert 立方体，并去除了一个点。我们还证明，对于 n ≤ 2 ，GH (B n ) 同胚于 Hilbert 立方体。