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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Intersecting unions of maximal convex sets


Author: Marilyn Breen
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
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0319046-0
Keywords: Maximal convex subsets, unions of convex sets
Article copyright: © Copyright 1973 American Mathematical Society