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Helly-Type Theorems for Homothets of Planar Convex Curves

Konrad J. Swanepoel
Proceedings of the American Mathematical Society
Vol. 131, No. 3 (Mar., 2003), pp. 921-932
Stable URL: http://www.jstor.org/stable/1194496
Page Count: 12
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Helly-Type Theorems for Homothets of Planar Convex Curves
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Abstract

Helly's theorem implies that if S is a finite collection of (positive) homothets of a planar convex body B, any three having non-empty intersection, then S has non-empty intersection. We show that for collections S of homothets (including translates) of the boundary ∂ B, if any four curves in S have non-empty intersection, then 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 ∂ B, then S can be covered by a translate [homothet] of ∂ B. These results are best possible in general.

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