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Proceedings of the American Mathematical Society

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Starshaped unions and nonempty intersections of convex sets in $ {\bf R}\sp d$


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 108 (1990), 817-820
MSC: Primary 52A35; Secondary 52A30
DOI: https://doi.org/10.1090/S0002-9939-1990-0990413-5
MathSciNet review: 990413
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{G}$ be a nonempty family of compact convex sets in $ {R^d},d \geq 1$. Then every subfamily of $ \mathcal{G}$ consisting of $ d + 1$ or fewer sets has a starshaped union if and only if $ \cap \{ G:G{\text{ in }}\mathcal{G}\} \ne \emptyset $.


References [Enhancements On Off] (What's this?)

  • [1] Ludwig Danzer, Branko Grünbaum, and Victor Klee, Helly's theorem and its relatives, Convexity, Proc. Sympos. Pure Math., Vol. 7, Amer. Math. Soc., Providence, RI, 1962, pp. 101-180.
  • [2] V. L. Klee Jr., On certain intersection properties of convex sets, Canadian J. Math. 3 (1951), 272–275. MR 0042726
  • [3] Krzysztof Kołodziejczyk, On starshapedness of the union of closed sets in 𝑅ⁿ, Colloq. Math. 53 (1987), no. 2, 193–197. MR 924062
  • [4] M. Krasnosselsky, Sur un critère pour qu’un domaine soit étoilé, Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 309–310 (Russian, with French summary). MR 0020248
  • [5] Steven R. Lay, Convex sets and their applications, John Wiley & Sons, Inc., New York, 1982. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 655598
  • [6] Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0990413-5
Article copyright: © Copyright 1990 American Mathematical Society