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.
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© 2001 Springer-Verlag Berlin Heidelberg
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Páles, Z. (2001). Separation Theorems for Convex Sets and Convex Functions with Invariance Properties. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, JP. (eds) Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems, vol 502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56645-5_20
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DOI: https://doi.org/10.1007/978-3-642-56645-5_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41806-1
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