Slice-continuous sets in reflexive Banach spaces: convex constrained optimization and strict convex separation
Continuous sets (i.e., sets for which the support function is continuous) play a crucial role for the existence of global minimizers of continuous convex functions on closed convex feasible sets, but also for the strict separation of disjoint closed convex sets. To extend known results of finite dimensional spaces to reflexive Banach spaces, the authors generalize the notion of continuous sets and introduce the notion of slice-continuous sets. A closed convex set \(C\) is called slice-continuous if \(C\cap L\) is continuous with respect to \(L\) for every closed linear manifold \(L\) which meets \(C\). Many equivalent conditions for this notion are presented. In the first main theorem, it is shown that a nonconstant continuous convex function \(\Phi\) on a reflexive Banach space \(X\) (which attains its infimum on \(X\)) attains its infimum on each closed convex subset of \(X\) if and only if every nonempty level set of \(\Phi\) is a slice-continuous set. In the second main theorem, it is pointed out that a nonempty closed convex proper set \(C\subset X\) can be strictly separated from every other disjoint nonempty closed convex set if and only if \(C\) is slice-continuous.
scientific article; zbMATH DE number 2183825
scientific article