The Gromov-Hausdorff hyperspace of a Euclidean space

Abstract

We investigate the hyperspace GH(Rn) 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(Rn) is homeomorphic to the orbit space 2Rn/E(n) of the hyperspace 2Rn of all non-empty compact subsets of a Euclidean space Rn equipped with the Hausdorff metric and the natural action of the Euclidean group E(n). This is further applied to prove that 2Rn/E(n) is homeomorphic to the open cone OCone(Ch(Bn)/O(n)), where Ch(Bn) stands for the set of all A∈2Rn for which the closed Euclidean unit ball Bn is the least circumscribed ball (the Chebyshev ball). These results lead to determine the complete topological structure of GH(Rn) for n≤2, namely, we prove that GH(Rn) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n≤2, GH(Bn) is homeomorphic to the Hilbert cube.