Classical and continuous Gromov-Hausdorff distances

TL;DR

This paper explores modifications of the Gromov-Hausdorff distance by imposing semicontinuity conditions on correspondences, analyzing their effects and equivalences with the classical metric under various boundedness assumptions.

Contribution

It introduces semicontinuity constraints into the Gromov-Hausdorff distance framework and characterizes when these modified distances coincide with the classical one.

Findings

Lower semicontinuity yields the classical Gromov-Hausdorff distance.

Upper semicontinuity matches the classical distance for totally bounded or boundedly compact spaces.

The study clarifies the role of semicontinuity in the stability of Gromov-Hausdorff distances.

Abstract

Starting from the definition of the Gromov-Hausdorff distance via distortion of correspondences, we add the requirement of semicontinuity of each correspondence and its inverse. It turns out that in the case of lower semicontinuity we obtain the same classical Gromov-Hausdorff distance, while for upper semicontinuity we are able to prove coincidence with the classical one only in cases where the spaces are either totally bounded or boundedly compact.