Approximating Quasiconvex Functions with Strictly Quasiconvex Ones in Banach Space

In this paper we show how to approximate a quasiconvex function with a sequence of strictly quasiconvex functions in a reflexive Banach space X. An important role in our approximation procedure is played by a real valued convex function defined on X, and parameterized by a pair of closed bounded convex sets, which is a generalization of the classical Minkowski functional on X; for this reason, we investigate some of its properties. In particular, we prove the continuity of this map, seen as a function acting from a specific family of pairs of closed convex subsets of X, to the space of the real valued continuous functions on X. In the domain space we use the (bounded) Hausdorff topology, while the target space is endowed with the topology of the uniform convergence on bounded sets. Our results also need to approximate a closed convex set, in the sense of the bounded Hausdorff topology, with a sequence of strictly convex sets. The result particularizes to Hausdorff topology if the limit set is bounded.