A Planar 3-Convex Set is Indeed a Union of Six Convex Sets

Noa Nitzan*, Micha A. Perles

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageAmerican English
Pages (from-to)454-477
Number of pages24
JournalDiscrete and Computational Geometry
Issue number3
StatePublished - Apr 2013


  • Invisibility graph
  • Non-convexity
  • Seeing subset
  • Valentine's Theorem (57')
  • Visually independent


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