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Intersection Patterns of Families of Convex Sets

Published online by Cambridge University Press:  20 November 2018

M. Katchalski  and
A. Liu
M. Katchalski
Affiliation:
Technion, Haifa, Israel
A. Liu
Affiliation:
University of Alberta, Edmonton, Alberta
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In this paper, we study the intersection pattern of families of convex sets. Since we only consider finite families, we may assume that the sets are also compact.

As an example, we consider families of 5 convex sets in R2 such that every two intersect and no three intersect. One such family that comes immediately to mind is that of 5 lines in general position. However, this is not the only family which exhibits this intersection pattern. Fig. 1 shows a family of 3 lines (sides of the large triangle) and 2 triangles (inscribed in the large triangle) that has this property.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 4 , 01 August 1982 , pp. 921 - 931
Copyright
Copyright © Canadian Mathematical Society 1982

References

References>

1. Helly, E., Ûber Mengen Konverser Kurper mit gemeinschaftlichen Punkten, J. Deutsch. Math. Verein. 32 (1923), 175–176.Google Scholar
2. Katchalski, M. and Liu, A., A problem of geometry in Rn, Proc. Amer. Math. Soc. 75 (1979), 284–288.Google Scholar
3. Sperner, E., Ein Satz iiber Untermengen einer endlichen Menge, Math. 27 (1928), 544–548.Google Scholar
4. P., Turân, On an extremal problem in graph theory, Mat. Fiz. Lapok 48 (1941), 436–452.Google Scholar