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

Noa Nitzan; Micha A. Perles

doi:10.1007/s00454-012-9484-7

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

Noa Nitzan, Micha A. Perles

The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

Abstract

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 language	English
Pages (from-to)	454-477
Number of pages	24
Journal	Discrete and Computational Geometry
Volume	49
Issue number	3
DOIs	https://doi.org/10.1007/s00454-012-9484-7
State	Published - Apr 2013

Keywords

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

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s00454-012-9484-7

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abstract = "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.",

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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

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EP - 477

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 3

ER -

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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