On sets having finitely many points of local nonconvexity and propertyP m

1971 ◽  
Vol 10 (2) ◽  
pp. 196-209 ◽  
Author(s):  
Merle D. Guay ◽  
David C. Kay
Keyword(s):  
Local Nonconvexity

A Decomposition Theorem for m-Convex Sets in Rd

1976 ◽  
Vol 28 (5) ◽  
pp. 1051-1057
Author(s):  
Marilyn Breen
Keyword(s):  
Topological Space ◽  
Convex Sets ◽  
Decomposition Theorem ◽  
Linear Topological Space ◽  
Local Convexity ◽  
Line Segments ◽  
Local Nonconvexity ◽  
Locally Convex

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 S, at least one of the line segments determined by these points lies in S. A point x in S is said to be a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z Є N ⌒ S, then [y, z] ⊆ S. If S fails to be locally convex at some point a in S, then q is called a point of local nonconvexity (lnc point) of S.

Intersections of m-Convex Sets

1975 ◽  
Vol 27 (6) ◽  
pp. 1384-1391 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Topological Space ◽  
Convex Sets ◽  
Linear Topological Space ◽  
Local Convexity ◽  
Line Segments ◽  
Local Nonconvexity ◽  
Locally Convex

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.

Two Cells with N Points of Local Nonconvexity

1971 ◽  
Vol 27 (2) ◽  
pp. 331
Author(s):  
Nick M. Stavrakas ◽  
W. R. Hare ◽  
J. W. Kenelly
Keyword(s):  
Local Nonconvexity

Ln Sets and the Closures of Open Connected Sets

1975 ◽  
Vol 27 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nick M. Stavrakas
Keyword(s):  
Convex Sets ◽  
Connected Subset ◽  
Local Connectivity ◽  
Connected Sets ◽  
Connected Set ◽  
Local Nonconvexity ◽  
Connectivity Property ◽  
Number Of Authors

F. A. Valentine in [4] proved the following two theorems.THEOREM 1. Let S be a closed connected subset of Rd which has at most n points of local nonconvexity. Then S is an Ln+i set.THEOREM 2. Let S be a closed connected subset of Rd whose points of local nonconvexity are decomposable into n closed convex sets. Then S is an L2n+i set.These results have been extended by a number of authors, but always with stronger hypothesis. (See [1] and [2].) Using a minimal arc technique, new pr∞fs of Theorems 1 and 2 were given in [3].Valentine remarks in [4] that Theorem 2 might be improved in the case that 5 is the closure of an open connected set. The goal of this paper is to give such an improvement for sets satisfying a particular local connectivity property.

Essential and inessential points of local nonconvexity

1972 ◽  
Vol 12 (4) ◽  
pp. 347-355
Author(s):  
Marilyn Breen
Keyword(s):  
Local Nonconvexity
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