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Semiconvex geometry

Part of: General convexity Categories and theories

Published online by Cambridge University Press:  09 April 2009

Joe Flood
Joe Flood
Affiliation:
CSIRO Division of Building Research, P.O. Box 56, Highett, Victoria 3190, Australia
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Abstract

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Semiconvex sets are objects in the algebraic variety generated by convex subsets of real linear spaces. It is shown that the fundamental notions of convex geometry may be derived from an entirely algebraic approach, and that conceptual advantages result from applying notions derived from algebra, such as ideals, to convex sets. Some structural decomposition results for semiconvex sets are obtained. An algebraic proof of the algebraic Hahn-Banach theorem is presented.

MSC classification

Secondary: 52A01: Axiomatic and generalized convexity 18C05: Equational categories
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 4 , April 1981 , pp. 496 - 510
Copyright
Copyright © Australian Mathematical Society 1981

References

Bourbaki, N. (1953), Elements de mathematique, Livre 5, Espaces vectoriels topologiques Vol. I (Act. Sci. et Ind.).Google Scholar
Grätzer, G. (1968), Universal algebra (Van Nostrand, Princeton, N. J.).Google Scholar
Mac Lane, S. (1971), Categories for the working mathematician (Graduate Texts in Mathematics, 5, Springer-Verlag, Berlin).Google Scholar
Semadeni, Z. (1971), Banach spaces of continuous functions I (Monografie Matematyczne, 55. Warsaw).Google Scholar
Swirszck, T. (1973), Monadic functors and categories of convex sets, Preprint No. 70. (Proc. Inst. Math. Pol. Acad. Sci., Warsaw).Google Scholar