  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.
