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 -