Dual Characterizations of Set Containments with Strict Convex Inequalities

Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a …nite union of convex sets (i.e. the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classi…cation problems where they are used to describe the containments as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments. Key words: Set containment, convex functions, semi-in…nite systems, existence theorems, dual cones, conjugacy.