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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Property $ P\sb{3}$ and the union of two convex sets


Author: E. O. Buchman
Journal: Proc. Amer. Math. Soc. 25 (1970), 642-645
MSC: Primary 52.30
MathSciNet review: 0259750
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Abstract: A set $ S$ in a linear space is said to have the three-point convexity property $ {P_3}$ iff for each triple of points $ x,\;y,\;z$ of $ S$, at least one of the segments $ xy,\;xz,\;yz$ is a subset of $ S$. It is proved that if $ S$ is a compact set in Euclidean space of dimension at least three with at least one point interior to its convex kernel and if the set of points of local nonconvexity of $ S$ is interior to its convex hull, then $ S$ has property $ {P_3}$ iff it is the union of two convex sets.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0259750-3
PII: S 0002-9939(1970)0259750-3
Article copyright: © Copyright 1970 American Mathematical Society