Elsevier

Advances in Mathematics

Volume 363, 25 March 2020, 106977
Advances in Mathematics

The Gromov-Hausdorff hyperspace of a Euclidean space

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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 n1, 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 A2Rn 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 n2, namely, we prove that GH(Rn) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n2, GH(Bn) is homeomorphic to the Hilbert cube.

MSC

51F99
57S20
57S25
57N20
54B20
54C55

Keywords

Gromov-Hausdorff metric
Urysohn universal metric space
Orbit space
Hyperspace
Q-manifold

Cited by (0)

The author was supported in part by grants IN-101420 from PAPIIT (UNAM) and A1-S-7897 from CONACYT.