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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

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