Gap and Excess Functionals and Weak Topologies

Abstract

If x is a fixed point of a metric space <X,d>,the distance functional d(x,·) : CL(X) → [0,+∞) is both a gap functional and an excess functional with respect to the fixed set argument {x}. In this chapter, we consider in a systematic way topologies generated by prescribed families of gap functionals and/or excess functionals, varying metrics as well as the fixed set argument for the functionals. This program not only subsumes all the topologies we encountered in Chapter 2, but also provides new presentations for these topologies as well. Both the Attouch-Wets topology and the Hausdorff metric topology arise in this way, and in the convex case, the topology of pointwise convergence of support functionals for convex sets may be so characterized. Of particular interest from the point of view of operations on convex sets is the supremum of the Wijsman topology and the topology of pointwise convergence of support functionals. In the final section we consider the general problem of uniformization of hit-and-miss and proximal hit-and-miss topologies. We show that such a topology is uniformizable if and only if it arises as a weak topology determined by a family of infimal value functionals, of which the gap functional is a particular example.