Finite unions of convex sets
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- by J. F. Lawrence, W. R. Hare and John W. Kenelly PDF
- Proc. Amer. Math. Soc. 34 (1972), 225-228 Request permission
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
- 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 257879, DOI 10.1090/S0002-9939-1970-0257879-7 Richard L. McKinney, On unions of more than two convex sets, Notices Amer. Math. Soc. 17 (1970), 249. Abstract #672-575.
- Richard L. McKinney, On unions of two convex sets, Canadian J. Math. 18 (1966), 883–886. MR 202049, DOI 10.4153/CJM-1966-088-7
- F. A. Valentine, A three point convexity property, Pacific J. Math. 7 (1957), 1227–1235. MR 99632
Additional Information
- © Copyright 1972 American Mathematical Society
- 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