Helly-type theorems for homothets of planar convex curves
Konrad J. Swanepoel · Proceedings of the American Mathematical Society · 2002
Helly’s theorem implies that if S \boldsymbol {\mathcal {S}} is a finite collection of (positive) homothets of a planar convex body B B , any three having non-empty intersection, then S \boldsymbol {\mathcal {S}} has non-empty intersection. We show that for collections S \boldsymbol {\mathcal {S}} of homothets (including translates) of the boundary ∂ B \partial B , if any four curves in S \boldsymbol {\mathcal {S}} have non-empty intersection, then S \boldsymbol {\mathcal {S}} has non-empty intersection. We prove the following dual version: If any four points of a finite set S S in the plane can be covered by a translate [homothet] of ∂ B \partial B , then S S can be covered by a translate [homothet] of ∂ B \partial B . These results are best possible in general.