Hausdorff Sampling of Closed Sets into a Boundedly Compact Space

Abstract

Our theory of Hausdorff discretization has been given in the following framework [10]. Assume an arbitrary metric space (E,d) (E can be a Euclidean space)and a nonvoid proper subspace D of E (the discrete space)such that:(1)D is boundedly finite,that is every bounded subset of D is finite,and (2)the distance frompoints of E to D is bounded; we call this bound the covering radius,it is a measure of the resolution of D For every nonvoid compact subset K of E any nonvoid finite subset S of D such that the Hausdorff distance between S and K is minimal is called a Hausdorff discretizing set (or Hausdorff discretization)of K; among such sets there is always a greatest one (w.r.t.inclusion),which we call the maximal Hausdorff discretization of K. The distance between a compact set and its Hausdorff discretizing sets is bounded by the covering radius,so that these discretizations converge to the original compact set (for the Hausdorff metric)when the resolution of D tends to zero. Here we generalize this theory in two ways. First,we relax condition (1)on D we assume simply that D is boundedly compact,that is every closed bounded subset of D is compact. Second,the set K to be discretized needs not be compact,but boundedly compact,or more generally closed (cfr.[15]in the particular case whereE =R ⁿand D =Z ⁿ).