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Maximal separation theorems for convex sets

1968; American Mathematical Society; Volume: 134; Issue: 1 Linguagem: Inglês

10.1090/s0002-9947-1968-0235457-9

1088-6850

Victor Klee,

Advanced Optimization Algorithms Research

Introduction.Throughout the paper E denotes a finite-dimensional Euclidean space.Attention is restricted to subsets of E though some of the results have infinite-dimensional analogues.Separation theorems for convex sets are basic in the theory of convexity and in the applications of convexity to other parts of mathematics.The standard separation theorem asserts that any two nonempty disjoint convex subsets of E are separated by a hyperplane; this involves a very weak type of separation.Other well-known separation theorems deal with compact or open convex sets and with rather strong types of separation.Several additional separation theorems have appeared in the literature [2], [4], [6], most of them directed at classes of closed convex sets (intersections of closed halfspaces).Such theorems are of interest in connection with systems of weak (S) linear inequalities.The main separation theorems of this paper are directed at classes of evenly convex sets (intersections of open halfspaces [1]) and are thus of interest in connection with systems of strong ( < ) linear inequalities or mixed systems of inequalities.Rockafellar's separation theorem [10] for partially polyhedral sets is extended to a wider class of sets.The attempt to obtain separation theorems under minimal hypotheses leads to the notion of a maximal separation theorem.Eighteen maximal theorems are presented here, involving four different types of separation.Definitions and preliminaries.Let us begin by defining the various types of separation to be considered.A set X is said to be separated from a set F by a hyperplane 77 provided that X lies in one of the closed halfspaces bounded by 77 and Y lies in the other.In contrast to the other types of separation, this does not require disjointness of X and Y; however, we shall be concerned only with disjoint sets.The set X is nicely separated from F by 77 (X | Y by 77) provided that the separating hyperplane 77 is disjoint from X or from Y (without specifying which), openly separated from F by 77 (X-\ Y by 77) provided that 77 is disjoint from X, and closedly separated from Y by 77 (X\ ■ Y by 77) provided that 77 is disjoint from Y.Thus X is openly or closedly separated from F by 77 according as X lies in an open or a closed halfspace disjoint from Y and bounded by 77.The set X is strictly separated from F by 77 (X-\ ■ Y by 77) provided that the separating hyperplane 77 is