$M$-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.