Extremal Systems of Convex Sets with Applications to Convex Calculus in Vector Spaces

In this paper we introduce and study the concept of set extremality for systems of convex sets in vector spaces without topological structures. Characterizations of the extremal systems of sets are obtained in the form of the convex extremal principle, which is shown to be equivalent to convex separation under certain qualification conditions expressed via algebraic cores. The obtained results are applied via a variational geometric approach to deriving enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued convex functions including the optimal value ones. These rules of the equality type are established under refined qualification conditions in terms of algebraic cores in arbitrary vector spaces. Our new developments partially answer the question on how far we can go with set-valued and convex analysis without any topological structure on the underlying spaces.