The Gromov-Hausdorff Metric on the Space of Compact Metric Spaces is Strictly Intrinsic

TL;DR

This paper proves that the Gromov-Hausdorff metric space of compact metric spaces is geodesic, meaning any two spaces can be connected by a shortest path, with explicit constructions for finite spaces.

Contribution

It establishes that the Gromov-Hausdorff space is strictly intrinsic and provides explicit geodesics for finite metric spaces.

Findings

The Gromov-Hausdorff space is a geodesic metric space.

Explicit geodesics are constructed for finite metric spaces.

The space is strictly intrinsic, ensuring shortest paths between any two points.

Abstract

It is proved that the Gromov-Hausdorff metric on the space of compact metric spaces considered up to an isometry is strictly intrinsic, i.e., the corresponding metric space is geodesic. In other words, each two points of this space (each two compact metric spaces) can be connected by a geodesic. For finite metric spaces a geodesic is constructed explicitly.