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Intersections of m-Convex Sets

Published online by Cambridge University Press:  20 November 2018

Marilyn Breen
Marilyn Breen*
Affiliation:
University of Oklahoma, Norman, Oklahoma
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Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of line segments determined by these points lies in S. A point x in S is called a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z ∈ N⋂ S, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 6 , 01 December 1975 , pp. 1384 - 1391
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Breen, Marilyn, Points of local nonconvexity and finite unions of convex sets, Can. J. Math. 27 (1975), 376–383.Google Scholar
2. Breen, Marilyn and Kay, David C., General decomposition theorems for m-convex sets in the plane, submitted to Israel J. Math.Google Scholar
3. Hare, W. R. and Kenelly, John W., Intersections of maximal starshaped sets, Proc. Amer. Math. Soc. 19 (1968), 1299–1302.Google Scholar
4. Kay, David C. and Guay, Merle D., Convexity and a certain property Pm, Israel J. Math. 8 (1970), 39–52.Google Scholar
5. Sparks, Arthur G., Intersections of maximal Ln sets, Proc. Amer. Math. Soc. 24 (1970), 245–250.Google Scholar
6. Stamey, W. L. and Marr, J. M., Unions of two convex sets, Can. J. Math. 15 (1963), 152–156.Google Scholar
7. Tattersall, J. J., On the intersection of maximal m-convex subsets, Israel J. Math. 16 (1963), 300–305.Google Scholar
8. Valentine, F. A., Local convexity and Ln sets, Proc. Amer. Math. Soc. 16 (1965), 1305–1310.Google Scholar
9. Valentine, F. A., A three point convexity property, Pacific J. Math. 7 (1957), 1227–1235.Google Scholar