Helly-type theorems for homothets of planar convex curves |
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| Authors: | Konrad J Swanepoel |
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| Institution: | Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa |
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| Abstract: | Helly's theorem implies that if  is a finite collection of (positive) homothets of a planar convex body  , any three having non-empty intersection, then  has non-empty intersection. We show that for collections  of homothets (including translates) of the boundary  , if any four curves in  have non-empty intersection, then  has non-empty intersection. We prove the following dual version: If any four points of a finite set  in the plane can be covered by a translate homothet] of  , then  can be covered by a translate homothet] of  . These results are best possible in general. |
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| Keywords: | Helly-type theorem convex curves congruence index congruence indices |
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