Sets which can be extended to -convex sets

Author:
Marilyn Breen

Journal:
Proc. Amer. Math. Soc. **62** (1977), 124-128

MSC:
Primary 52A20

MathSciNet review:
0430962

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Abstract: Let *S* be a compact set in . Then lies in an *m*-convex subset of *S* if and only if every finite subset of lies in an *m*-convex subset of *S*. For *S* a closed set in and , let a polytope in *S* having vertex set in . If for every three members of , at least one of the corresponding convex hulls

*S*, then lies in a 3-convex subset of

*S*. An analogous result holds for

*m*-convex sets provided ker .

**[1]**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****[2]**J. F. Lawrence, W. R. Hare Jr., and John W. Kenelly,*Finite unions of convex sets*, Proc. Amer. Math. Soc.**34**(1972), 225–228. MR**0291952**, 10.1090/S0002-9939-1972-0291952-4**[3]**William L. Stamey and J. M. Marr,*Unions of two convex sets*, Canad. J. Math.**15**(1963), 152–156. MR**0145415****[4]**Frederick A. Valentine,*Convex sets*, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR**0170264****[5]**F. A. Valentine,*A three point convexity property*, Pacific J. Math.**7**(1957), 1227–1235. MR**0099632**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1977-0430962-5

Keywords:
*m*-convex sets

Article copyright:
© Copyright 1977
American Mathematical Society