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Proceedings of the American Mathematical Society

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



A bound for decompositions of $ m$-convex sets whose LNC points lie in a hyperplane

Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 72 (1978), 159-162
MSC: Primary 52A20
MathSciNet review: 0640747
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Abstract: A set S in $ {R^d}$ is said to be m-convex, $ m \geqslant 2$, if and only if for every m points in S, at least one of the line segments determined by these points lies in S. Let S denote a closed m-convex set in $ {R^d}$, and assume that the set of lnc points of S lies in a hyperplane. Then S is a union of $ f(m)$ or fewer convex sets, where f is defined inductively as follows: $ f(2) = 1,f(3) = 2$, and $ f(m) = f(m - 2) + 3$ for $ m \geqslant 4$. Moreover, for $ d \geqslant 3$, an example reveals that the best bound is no lower than $ g(m)$, where $ g(m) = f(m)$ for $ 2 \leqslant m \leqslant 5$ and for $ m = 7$, and $ g(m) = g(m - 3) + 4$ otherwise.

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Article copyright: © Copyright 1978 American Mathematical Society