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Intersection Patterns of Families of Convex Sets

  • M. Katchalski (a1) and A. Liu (a2)

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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 R 2 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.

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References

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1. Helly, E., Ûber Mengen Konverser Kurper mit gemeinschaftlichen Punkten, J. Deutsch. Math. Verein. 32 (1923), 175176.
2. Katchalski, M. and Liu, A., A problem of geometry in Rn, Proc. Amer. Math. Soc. 75 (1979), 284288.
3. Sperner, E., Ein Satz iiber Untermengen einer endlichen Menge, Math. 27 (1928), 544548.
4. P., Turân, On an extremal problem in graph theory, Mat. Fiz. Lapok 48 (1941), 436452.
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Intersection Patterns of Families of Convex Sets

  • M. Katchalski (a1) and A. Liu (a2)

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