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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Helly-type theorems for homothets of planar convex curves
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by Konrad J. Swanepoel PDF
Proc. Amer. Math. Soc. 131 (2003), 921-932 Request permission

Abstract:

Helly’s theorem implies that if $\boldsymbol {\mathcal {S}}$ is a finite collection of (positive) homothets of a planar convex body $B$, any three having non-empty intersection, then $\boldsymbol {\mathcal {S}}$ has non-empty intersection. We show that for collections $\boldsymbol {\mathcal {S}}$ of homothets (including translates) of the boundary $\partial B$, if any four curves in $\boldsymbol {\mathcal {S}}$ have non-empty intersection, then $\boldsymbol {\mathcal {S}}$ has non-empty intersection. We prove the following dual version: If any four points of a finite set $S$ in the plane can be covered by a translate [homothet] of $\partial B$, then $S$ can be covered by a translate [homothet] of $\partial B$. These results are best possible in general.
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Additional Information
  • Konrad J. Swanepoel
  • Affiliation: Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa
  • Email: swanekj@unisa.ac.za
  • Received by editor(s): October 12, 2000
  • Received by editor(s) in revised form: October 23, 2001
  • Published electronically: July 17, 2002
  • Communicated by: John R. Stembridge
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 921-932
  • MSC (2000): Primary 52A23; Secondary 52A10
  • DOI: https://doi.org/10.1090/S0002-9939-02-06722-9
  • MathSciNet review: 1937431