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An Internal Solution to the Problem of Linearization of a Convexity Space

Published online by Cambridge University Press:  20 November 2018

D. A. Szafron  and
J. H. Weston
D. A. Szafron
Affiliation:
Department of Mathematics, University of ReginaRegina, Saskatchewan, CanadaS4S OA2
J. H. Weston
Affiliation:
Department of Mathematics, University of ReginaRegina, Saskatchewan, CanadaS4S OA2
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Following Kay and Womble [2] an abstract convexity structure on a set X is a collection ξ of subsets of X which includes the empty set, X and is closed under arbitrary intersections. One of the natural problems that arises in convexity structures is to give necessary and sufficient conditions for the existance of a linear structure on X such that the collection of all convex sets in the resulting linear space is precisely ξ. An associated problem is to consider a set with a convexity structure and a topology and find necessary and sufficient conditions for the existance of a linear structure on X such that X becomes a linear topological space with again ξ the collection of convex sets.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 4 , 01 December 1976 , pp. 487 - 494
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Guay, M. D. and Naimpally, S. A., Characterization of a convex subspace of a linear topological space, Proc. Japan Acad, (to appear).Google Scholar
2. Kay, D. C. and Womble, E. W., Axiomatic convexity theory and the relationships between Caratheadory, Helly, and Radon numbers, Pacific J. Math., 15 (1971), 65–76.Google Scholar
3. Mah, D., Naimpally, S. A., and Whitefield, J. H. M., Linearization of a convexity structure, J. London Math. Soc, (to appear).Google Scholar
4. Moreman, Douglas Convexly topological space, (preprint).Google Scholar