Separation Theorems for Convex Sets and Convex Functions with Invariance Properties

Abstract

In this paper, a generalization of Stone’s celebrated separation theorem is offered. It is shown that if the given disjoint convex sets are invariant with respect to a commuting family of affine transformations, then they can be separated by complementary convex sets enjoying the same invariance properties. The recession cone of the separating sets can also be nonsmaller than that of the data. As applications, we investigate the separability of affine invariant convex sets. It turns out that the separating affine function inherits invariance properties from the data. The results obtained generalize the Hahn-Banach and the Dubovitskii-Milyutin separation theorems. Sandwich theorems are also considered for convex-concave and for sublinear-superlinear pairs of functions admitting further invariance properties. In this way, the Hahn-Banach extension theorem can also be generalized.2000 Mathematics Subject ClassificationPrimary 46A22Secondary 52A05Keywords and phrasesSeparation of convex setsStone’s theoremaffine transformationinvariant convex setseparation by hyperplanesDubovitskii-Milyutinseparation theoremsandwich theorem