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Points of Local Nonconvexity and Finite Unions of Convex Sets

Published online by Cambridge University Press:  20 November 2018

Marilyn Breen
Marilyn Breen*
Affiliation:
The University of Oklahoma, Norman, Oklahoma
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Let 5 be a subset of Rd. A point x in 5 is a point of local convexity of S if and only if there is some neighborhood U of x such that, if y, z ϵ 5 ⌒ U, then [y, z] ⊆ S. If S fails to be locally convex at some point q in 5, then q is called a point of local nonconvexity (lnc point) of S.

Several interesting properties are known about sets whose lnc points Q may be decomposed into n convex sets. For S closed, connected, S ∼ Q connected, and Q having cardinality n, Guay and Kay [2] have proved that S is expressible as a union of n + 1 or fewer closed convex sets (and their result is valid in a locally convex topological vector space).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 376 - 383
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Breen, Marilyn, Essential and inessential points of local nonconvexity, Israel J. Math. 12 (1972), 347–355.Google Scholar
2. Guay, Merle D. and Kay, David C., On sets having finitely many points of local nonconvexity and property Pm, Israel J. Math. 10 (1971), 196–209.Google Scholar
3. Stavrakas, Nick M., A generalization of Tietze's theorem on convex sets in R3, Proc. Amer. Math. Soc. 40 (1973), 565–567.Google Scholar
4. Stavrakas, Nick M., On the polygonal connectivity of polyhedra and the closures of open connected sets, Bull. Amer. Math. Soc. 79 (1973), 403–406.Google Scholar
5. Tietze, H., Ùber Konvexheit im kleinen und im grossen und iiber gewisse den Punkten einer Menge zugeordnete Dimensionszahlen, Math. Z. 28 (1928), 697–707.Google Scholar
6. Valentine, F. A., Convex sets (McGraw-Hill, New York, 1964).Google Scholar
7. Valentine, F. A., Local convexity and Ln sets, Proc. Amer. Math. Soc. 16 (1965), 1305–1310.Google Scholar