The authors present a new approach, concentrating on the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. They use a proper version of convex separation theorem, which holds in any vector space and is formulated via algebraic core conditions instead of the conventional interiority assumptions in topological settings. They show also that a simple ‘extreme’ version of this result is equivalent the Hahn Banach theorem in vector spaces; this allows them to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. They also obtain a precise formula for computing the subdifferential of optimal value functions associated with convex problems of parametric optimization in vector spaces. They emphasize that functions of this type play a crucial role in many aspects of convex optimization and its applications.