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arXiv · 1804.08430

Convexity of Balls in Gromov--Hausdorff Space

Daria P. Klibus
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Abstract

In this paper we study the space $\mathcal{M}$ of all nonempty compact metric spaces considered up to isometry, equipped with the Gromov--Hausdorff distance. We show that each ball in $\mathcal{M}$ 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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Daria P. Klibus. 2018-04-19. Convexity of Balls in Gromov--Hausdorff Space. https://arxiv.org/abs/1804.08430

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