Abstract
We prove that (i) a familyF of at leastn+3 spheres inE n has nonempty intersection if eachn+1 spheres ofF have nonempty intersection, and (ii) if a familyF of spheres inE n has nonempty intersection, then there existn+1 or fewer spheres inF whose intersection coincides with the intersection of all spheres ofF.
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L. Danzer, B. Grünbaum, and V. Klee, Helly's theorem and its relatives,Proceedings of Symposia in Pure Mathematics, Vol. 7, American Mathematical Society, Providence, RI, 1963.
E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten,Jahresber. Deutsch. Math.-Verein. 32 (1923), 175–176.
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Dedicated to Professor Itiro Tamura on his 60th birthday
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Maehara, H. Helly-type theorems for spheres. Discrete Comput Geom 4, 279–285 (1989). https://doi.org/10.1007/BF02187730
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DOI: https://doi.org/10.1007/BF02187730