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

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.