Separation of convex sets by extreme hyperplanes

The problem of separation of convex sets by extreme hyperplanes (functionals) in normed linear spaces is examined. The concept of a bar (a closed set of special form) is introduced; it is shown that a bar is characterized by the property that any point not lying in it can be separated from it by an extreme hyperplane. In two-dimensional spaces, in spaces with strictly convex dual, and in the space of continuous functions, any two bars are extremely separated. This property is shown to fail in the space of summable functions. A number of examples and generalizations are given. © 2013 Springer Science+Business Media New York.