TY - JOUR
T1 - A Planar 3-Convex Set is Indeed a Union of Six Convex Sets
AU - Nitzan, Noa
AU - Perles, Micha A.
PY - 2013/4
Y1 - 2013/4
N2 - Suppose S is a planar set. Two points a,b in S see each other via S if [a,b] is included in S. F. Valentine proved in 1957 that if S is closed, and if for every three points of S, at least two see each other via S, then S is a union of three convex sets. The pentagonal star shows that the number three is the best possible. We drop the condition that S is closed and show that S is a union of (at most) six convex sets. The number six is best possible.
AB - Suppose S is a planar set. Two points a,b in S see each other via S if [a,b] is included in S. F. Valentine proved in 1957 that if S is closed, and if for every three points of S, at least two see each other via S, then S is a union of three convex sets. The pentagonal star shows that the number three is the best possible. We drop the condition that S is closed and show that S is a union of (at most) six convex sets. The number six is best possible.
KW - Invisibility graph
KW - Non-convexity
KW - Seeing subset
KW - Valentine's Theorem (57')
KW - Visually independent
UR - http://www.scopus.com/inward/record.url?scp=84876455499&partnerID=8YFLogxK
U2 - 10.1007/s00454-012-9484-7
DO - 10.1007/s00454-012-9484-7
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AN - SCOPUS:84876455499
SN - 0179-5376
VL - 49
SP - 454
EP - 477
JO - Discrete and Computational Geometry
JF - Discrete and Computational Geometry
IS - 3
ER -