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Proceedings of the American Mathematical Society

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Helly-type theorems for homothets of planar convex curves


Author: Konrad J. Swanepoel
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
Published electronically: July 17, 2002
MathSciNet review: 1937431
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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

Keywords: Helly-type theorem, convex curves, congruence index, congruence indices
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
Article copyright: © Copyright 2002 American Mathematical Society