The Gromov–Hausdorff metric on the space of compact metric spaces is strictly intrinsic

Ivanov, A O; Николаева, Н. А.; Tuzhilin, A. A.

doi:10.1134/s0001434616110298

Cited by 27 publications

(13 citation statements)

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“…Ivanov, Nikolaeva, and Tuzhilin [9] proved that (M, GH) is a geodesic space by showing the existence of the mid-point of all two points of M. Klibus [11] proved that the closed ball in the Gromov-Hausdorff space centered at the one-point metric space is a geodesic space. Chowdhury and Mémoli [2] constructed an explicit geodesic in (M, GH) using an optimal closed correspondence (see also [8]).…”

Section: Introductionmentioning

confidence: 99%

Branching geodesics of the Gromov--Hausdorff distance

Ishiki¹

2021

Preprint

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In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov-Hausdorff distance. We then construct branching geodesics of the Gromov-Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov-Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.

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Section: Introductionmentioning

confidence: 99%

Branching geodesics of the Gromov--Hausdorff distance

Ishiki¹

2021

Preprint

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“…Ivanov, N.K. Nikolaeva and A.A. Tuzhilin showed that the Gromov-Hausdorff metric is strictly intrinsic [4].…”

Section: Introductionmentioning

confidence: 99%

Convexity of a Ball in the Gromov–Hausdorff Space

Klibus

2018

Moscow Univ. Math. Bull.

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In this paper we study the space M of all nonempty compact metric spaces considered up to isometry, equipped with the Gromov-Hausdorff distance. We show that each ball in M with center at the one-point space is convex in the weak sense, i.e., every two points of such a ball can be joined by a shortest curve that belongs to this ball; however, such a ball is not convex in the strong sense: it is not true that every shortest curve joining the points of the ball belongs to this ball. We also show that a ball of sufficiently small radius with center at a space of general position is convex in the weak sense.

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“…It is shown that each boundary set consisting of finite metric spaces only can be connected by a Steiner minimal tree. In the general case, the authors have solved the Steiner problem for 2-point boundaries [1], where the problem is equivalent to the fact that the ambient space is geodesic. General case of more than 2 boundary points has resisted to the authors attempts based on the Gromov precompactness criterion.…”

Section: Introductionmentioning

confidence: 99%