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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Property $P_{3}$ and the union of two convex sets


Author: E. O. Buchman
Journal: Proc. Amer. Math. Soc. 25 (1970), 642-645
MSC: Primary 52.30
DOI: https://doi.org/10.1090/S0002-9939-1970-0259750-3
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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Article copyright: © Copyright 1970 American Mathematical Society