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Intersections of translates of convex sets

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Arne Brøndsted
Arne Brøndsted
Affiliation:
Institute of Mathematics, University of Copenhagen, Denmark.
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Extract

A finite family (Ci)i ∊ I of at least two convex subsets of ℝn is said to have the intersection property provided that the set is non-empty for all families (ai)i ∊ I of points in ℝn. Previously, D. G. Larman [2, Theorem 3] has given a sufficient condition (which is not necessary) and an “almost” necessary condition (which is not sufficient) for to have the intersection property.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 24 , Issue 1 , June 1977 , pp. 122 - 129
Copyright
Copyright © University College London 1977

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References

1.Klee, V.. “Separation properties of convex cones ”, Proc. Amer. Math. Soc., 6 (1955), 313–318.CrossRefGoogle Scholar
2.Larman, D. G.. “On the inner aperture and intersections of convex sets ”, Pacific J. Math., 55 (1974), 219–232.CrossRefGoogle Scholar
3.Rockafellar, R. T.. Convex analysis (Princeton, New Jersey, 1970).CrossRefGoogle Scholar