CHARACTERIZATION OF COMPACT-SETS BY THEIR DILATION VOLUME

The uniqueness up to translation of the characterization of random com pact sets in Euclidean space by their dilation volumes is shown. The u nique correspondence is shown to be a homeomorphism with respect to su itable topologies. If set differences of volume zero are neglected, di lations by three-point sets are sufficient to determine a non-random c ompact set and the correspondence is again a homeomorphism with respec t to vague topologies.