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Discrete & Computational Geometry

, Volume 38, Issue 2, pp 389–397 | Cite as

The Empty Hexagon Theorem

  • Carlos M. Nicolas
Article

Abstract

Let P be a finite set of points in general position in the plane. Let C(P) be the convex hull of P and let CiP be the ith convex layer of P. A minimal convex set S of P is a convex subset of P such that every convex set of P ∩ C(S) different from S has cardinality strictly less than |S|. Our main theorem states that P contains an empty convex hexagon if C1P is minimal and C4P is not empty. Combined with the Erdos-Szekeres theorem, this result implies that every set P with sufficiently many points contains an empty convex hexagon, giving an affirmative answer to a question posed by Erdos in 1977.

Keywords

Convex Hull Convex Subset General Position Discrete Comput Geom Minimal Convex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of Mathematics, University of KentuckyLexington, KY 40506USA

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