Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T00:10:23.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Touching Convex Sets in the Plane

Published online by Cambridge University Press:  20 November 2018

Meir Katchalski  and
János Pach
Meir Katchalski
Affiliation:
Department of Mathematics, Technion-LLT. Haifa 32000 Israel
János Pach
Affiliation:
Courant Institute, New York University New York, New York 10012 U.S.A. and Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127 1364, Budapest Hungary
Rights & Permissions [Opens in a new window]

Abstract

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.

Two subsets of the Euclidean plane touch each other if they have a point in common and there is a straight line separating one from the other.

It is shown that there exists a positive constant c such that if are families of plane convex sets with for some k ≥ 1 and if every touches every then either contains k members having nonempty intersection.

Keywords

52A10convex sets
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 4 , 01 December 1994 , pp. 495 - 504
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Edelsbrunner, H., Algorithms in Combinatorial Geometry, Springer Verlag, Berlin, Heidelberg, New York, 1987.Google Scholar
2. Helly, E., Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jahresber. Deutsch. Math.- Verein. 32(1923), 175–176.Google Scholar
3. Kalai, G., Intersection patterns of convex sets, Israel J. Math. 48(1984), 161–174.Google Scholar
4. Kövári, P., Sós, V. T. and P. Turán, On a problem of K. Zarankiewicz, Colloq. Math. 3(1954), 50–57.Google Scholar
5. Katchalski, M. and Liu, A., Intersection patterns of families of convex sets, Canad. J. Math. 4(1982), 921– 931.Google Scholar
6. Larman, D., Matoušek, J., Pach, J. and Töröcsik, J., A Ramsey type result for planar convex sets, to appear.Google Scholar
7. Tietze, H., Über das Problem der Nachbargebiete im Raum, Monatsh. Math. 16(1905), 211–216.Google Scholar