Helly-type theorems for homothets of planar convex curves

Swanepoel, Konrad

doi:10.1090/S0002-9939-02-06722-9

Helly-type theorems for homothets of planar convex curves
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by Konrad J. Swanepoel

Proc. Amer. Math. Soc. 131 (2003), 921-932

DOI: https://doi.org/10.1090/S0002-9939-02-06722-9

Published electronically: July 17, 2002

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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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