Israel Journal of Mathematics

, Volume 113, Issue 1, pp 341–379 | Cite as

On visibility and covering by convex sets

  • Jiří Matoušek
  • Pavel Valtr


A setX⊆ℝ d isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX. Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn 6 convex sets. We improve their bound to 18n 3, and show a lower bound of order Ω(n 2). We also show that ifX⊆ℝ2 is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n 4+n 2λ) convex sets. Two other results onn-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4).


Extremal Point Convex Subset Chromatic Number Relative Interior Vertical Segment 
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Copyright information

© The Magnes Press 1999

Authors and Affiliations

  • Jiří Matoušek
    • 1
  • Pavel Valtr
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic

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