Convexity of Balls in Gromov--Hausdorff Space

TL;DR

This paper investigates the convexity properties of metric balls in the Gromov--Hausdorff space, revealing conditions under which they are weakly convex but not strongly convex.

Contribution

It establishes that balls centered at the one-point space are weakly convex but not strongly convex, and identifies conditions for weak convexity at other centers.

Findings

Balls centered at the one-point space are weakly convex.

Such balls are not strongly convex.

Small-radius balls at general position spaces are weakly convex.

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.