Point-assigned distance-like functions on non-compact geodesic spaces

TL;DR

This paper introduces a new class of distance-like functions on non-compact geodesic spaces, creating a pseudo-metric on compact subsets that relates to large-scale geometry and has stability properties.

Contribution

It defines and analyzes point-assigned distance-like functions, establishing their properties, stability, and relation to the space's large-scale geometry, with applications to representing other distance-like functions.

Findings

The pseudo-metric is less than the Hausdorff distance.

Level sets of the functions are compact.

The functions are stable under Gromov-Hausdorff topology.

Abstract

On a complete, connected, locally compact, non-compact geodesic space $(X,d)$, we assign each compact set a distance-like function. With the help of these functions, we obtain a pseudo-metric on the space of (non-empty) compact subsets of $X$ which is less than the Hausdorff distance. The quotient metric space is closely related to the large scale geometry of the ambient metric space. In particular, we study both extreme cases --the pseudo-metric either vanishes or equals the original distance. The compactness of the level sets as well as stability under the Gromov-Hausdorff topology of such dl-functions are also investigated. As an application, we also give a representation formula of any distance-like function in terms of the singleton-assigned distance-like functions defined here.