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 | References | Similar Articles | Additional Information

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