A bound for decompositions of -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 is said to be *m*-convex, , 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 , and assume that the set of lnc points of *S* lies in a hyperplane. Then *S* is a union of or fewer convex sets, where *f* is defined inductively as follows: , and for . Moreover, for , an example reveals that the best bound is no lower than , where for and for , and otherwise.

**[1]**Marilyn Breen,*𝑚-convex sets whose lnc points lie in a hyperplane*, J. London Math. Soc. (2)**16**(1977), no. 3, 529–535. MR**0461292****[2]**Marilyn Breen and David C. Kay,*General decomposition theorems for 𝑚-convex sets in the plane*, Israel J. Math.**24**(1976), no. 3-4, 217–233. MR**0417925****[3]**H. G. Eggleston,*A condition for a compact plane set to be a union of finitely many convex sets*, Proc. Cambridge Philos. Soc.**76**(1974), 61–66. MR**0343175****[4]**F. A. Valentine,*A three point convexity property*, Pacific J. Math.**7**(1957), 1227–1235. MR**0099632****[5]**F. A. Valentine,*Local convexity and 𝐿_{𝑛} sets*, Proc. Amer. Math. Soc.**16**(1965), 1305–1310. MR**0185510**, 10.1090/S0002-9939-1965-0185510-6

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DOI:
https://doi.org/10.1090/S0002-9939-1978-0640747-3

Article copyright:
© Copyright 1978
American Mathematical Society