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Finite unions of convex sets


Authors: J. F. Lawrence, W. R. Hare and John W. Kenelly
Journal: Proc. Amer. Math. Soc. 34 (1972), 225-228
MSC: Primary 52A05; Secondary 46A15
DOI: https://doi.org/10.1090/S0002-9939-1972-0291952-4
MathSciNet review: 0291952
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Abstract: In this paper it is shown that a set is the union of k convex subsets if and only if every finite subset of it is contained in some k convex subsets of it. This is a characterization of a set as the union of a finite number of convex sets by conditions on its finite subsets.

Also, a proof of McKinney's theorem for unions of two convex sets is given using similar methods.


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

  • [1] W. R. Hare, Jr. and John W. Kenelly, Sets expressible as unions of two convex sets, Proc. Amer. Math. Soc. 25 (1970), 379-380. MR 41 #2528. MR 0257879 (41:2528)
  • [2] Richard L. McKinney, On unions of more than two convex sets, Notices Amer. Math. Soc. 17 (1970), 249. Abstract #672-575.
  • [3] -, On unions of two convex sets, Canad. J. Math. 18 (1966), 883-886. MR 34 #1923. MR 0202049 (34:1923)
  • [4] F. A. Valentine, A three point convexity property, Pacific J. Math. 7 (1957), 1227-1235. MR 20 #6071. MR 0099632 (20:6071)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0291952-4
Keywords: Unions of convex sets, convex kernel
Article copyright: © Copyright 1972 American Mathematical Society

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