Intersecting unions of maximal convex sets
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- by Marilyn Breen PDF
- Proc. Amer. Math. Soc. 39 (1973), 587-590 Request permission
Abstract:
Hare and Kenelly have characterized the intersection of the maximal starshaped subsets of a set $S$, where $S$ is compact, simply connected and planar, and Sparks has solved the general problem for maximal ${L_n}$ sets. In this paper, a similar question is examined for unions of maximal convex sets: Let $S$ be a subset of ${R^2},\mathcal {C}$ the collection of all maximal convex subsets of $S$, and $\mathcal {N} = \{ A \cup B:A,B$ distinct members of $\mathcal {C}\}$. Then ${ \cap ^\mathcal {N}}$ is expressible as a union of three or fewer convex sets.References
- W. R. Hare Jr. and John W. Kenelly, Intersections of maximal starshaped sets, Proc. Amer. Math. Soc. 19 (1968), 1299–1302. MR 233283, DOI 10.1090/S0002-9939-1968-0233283-3
- Arthur G. Sparks, Intersections of maximal $L_{n}$ sets, Proc. Amer. Math. Soc. 24 (1970), 245–250. MR 253153, DOI 10.1090/S0002-9939-1970-0253153-3
- F. A. Valentine, A three point convexity property, Pacific J. Math. 7 (1957), 1227–1235. MR 99632
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 39 (1973), 587-590
- MSC: Primary 52A10
- DOI: https://doi.org/10.1090/S0002-9939-1973-0319046-0
- MathSciNet review: 0319046