Evenly Convex Sets: Linear Systems Containing Strict Inequalities

This chapter deals with linear systems with an arbitrary (possibly infinite) number of weak and/or strict inequalities and their solution sets, the so-called evenly convex sets, which can be seen as the two faces of a same coin. Section 1.1 provides different characterizations of evenly convex sets and shows that this class of sets enjoys most of the well-known properties of the subclass of closed convex sets. Since the intersection of evenly convex sets belongs to the same family, any set has an evenly convex hull. Section 1.2 is focussed on the operations with evenly convex hulls and their relationships with other hulls. Section 1.3 reviews different types of separation theorems involving evenly convex sets. Section 1.4 provides existence theorems for linear systems with strict inequalities and characterizations of the linear inequalities which are consequence of consistent systems (those systems with nonempty solution set), which allows us to tackle set containment problems involving evenly convex sets. Section 1.5 is aimed to study the so-called evenly linear semi-infinite programming problems (i.e., linear semi-infinite programming problems with strict inequalities). Finally, Sect. 1.6 describes applications to polarity (treated in a detailed way as it was the problem which inspired the concept of evenly convex set), semi-infinite games, approximate reasoning, optimality conditions in mathematical programming, and strict separation of families of sets.