Slice-continuous sets in reflexive Banach spaces: convex constrained optimization and strict convex separation

The concept of continuous set has been used in finite dimension by Gale and Klee and recently by Auslender and Coutat. Here, we introduce the notion of slice-continuous set in a reflexive Banach space and we show that the class of such sets can be viewed as a subclass of the class of continuous sets. Further, we prove that every nonconstant real-valued convex and continuous function, which has a global minima, attains its infimum on every nonempty convex and closed subset of a reflexive Banach space if and only if its nonempty level sets are slice-continuous. Thereafter, we provide a new separation property for closed convex sets, in terms of slice-continuity, and conclude this article by comments.