A bound for decompositions of $m$-convex sets whose LNC points lie in a hyperplane
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- by Marilyn Breen PDF
- Proc. Amer. Math. Soc. 72 (1978), 159-162 Request permission
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.References
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Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 72 (1978), 159-162
- MSC: Primary 52A20
- DOI: https://doi.org/10.1090/S0002-9939-1978-0640747-3
- MathSciNet review: 0640747