M-strongly convex subsets and their generating sets

For subsets of a Banach space the notions of a generating set M and an M-strongly convex set are introduced. The latter can be represented as the intersection of sets of the form M + x, which are translates of the generating set M. A generating set must satisfy a condition that ensures a special support principle, as shown in the paper. Using this support principle a new area of convex analysis is constructed enabling one to strengthen classical results of the type of the Caratheodory and Krein-Milman theorems. Various classes of generating sets are described and the properties of M-strongly convex sets are studied.