On strongly convex sets and strongly convex functions (Q1585388)

The basic notions of this paper are generating set and \(M\)-strongly convex set, which have grown from the axiomatic approach to the notion of convexity. A convex closed set \(M\) of a Banach space \(E\) is called a generating set if for any nonempty set \(A\) of the form \(A=\bigcap_{x\in X}(M+x)\) one can find a convex closed set \(B\subset E\) such that \(\overline{A+B}=M\). For a given generating set \(M\) a nonempty set of the above form is called an \(M\)-strongly convex set. The author obtains necessary and sufficient conditions for a set to be a generating one and presents classes of generating sets, operations with generating sets that preserve the generating property, general properties of \(M\)-strongly convex sets for an arbitrary generating set \(M\) and conditions for preservation of the \(M\)-strong convexity. There are also introduced and studied the concepts of the \(M\)-strongly convex hull, \(R\)-strongly extreme point and \(R\)-strongly exposed point of a set. One can find here generalizations of the Carathéodory theorem on a representation of convex hull of a set in \(R^n\) and the Krein-Mil'man theorem on extreme points of a compact set in \(R^n\). Moreoever there is presented a new class of Lipschitzian single-valued selectors of convex- and compact-valued multivalued mappings. The author studies the class of generating sets which are the epigraphs of certain convex functions. He defines the concepts of a generating function \(m\), an \(m\)-strongly convex function (generalization of the notion of the strongly convex function) and an epidifference of functions (based on the Minkowski-Pontryagin difference of epigraphs of functions). He obtains a criterion for a function \(m\) to be a generating one and conditions for \(m\)-strong convexity of a given function. The paper includes a lot of interesting examples.