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 | References | Similar Articles | Additional Information
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.
- J. M. Marr and W. L. Stamey, A three-point property, Amer. Math. Monthly 69 (1962), 22–25. MR 137031, DOI https://doi.org/10.2307/2312728
- Richard L. McKinney, On unions of two convex sets, Canadian J. Math. 18 (1966), 883–886. MR 202049, DOI https://doi.org/10.4153/CJM-1966-088-7
- F. A. Valentine, A three point convexity property, Pacific J. Math. 7 (1957), 1227–1235. MR 99632
- Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
- F. A. Valentine, The intersection of two convex surfaces and property $P_{3}$, Proc. Amer. Math. Soc. 9 (1958), 47–54. MR 99633, DOI https://doi.org/10.1090/S0002-9939-1958-0099633-X
- Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491
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© Copyright 1970
American Mathematical Society