Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T13:12:14.536Z Has data issue: false hasContentIssue false
  • English
  • Français

On Certain Intersection Properties ofConvex Sets

Published online by Cambridge University Press:  20 November 2018

V. L. Klee Jr.
V. L. Klee Jr.*
Affiliation:
University of Virginia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A collection of n + 1 convex subsets of a Euclidean space E will be called an n-set in E provided each n of the sets have a common interior point although the intersection of all n + 1 interiors is empty. It is well-known that if {C0,C1} is a 1-set, then C0 and C1 can be separated by a hyperplane.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 272 - 275
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Helly, E., Über Mengen konvexer Körper mit gemeinschafilichen Punkten, Jber. Deutschen Math. Verein., vol. 32 (1923), 175–176.Google Scholar
[2] Horn, Alfred, Some generalizations of Helly's theorem on convex sets, Bull. Amer. Math. Soc, vol. 55 (1949), 923–929.Google Scholar
[3] Klee, V. L., Jr., Convex sets in linear spaces, Dissertation (University of Virginia, 1949).Google Scholar
[4] Kakutani, Shizuo, Ein Beweis des Satzes von M. Eidelheit uber konvexe Mengen, Proc. Imp. Acad., Tokyo, vol. 13 (1937), 93–94.Google Scholar
[5] Whyburn, G. T., Analytic Topology, Amer. Math. Soc. Colloquium Publications, vol. 28, (New York, 1942).Google Scholar