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INFINITE PATHS WITH NO SMALL ANGLES

Part of: Discrete geometry

Published online by Cambridge University Press:  10 December 2009

Imre Bárány  and
Attila Pór
Imre Bárány
Affiliation:
Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K. (email: [email protected])
Attila Pór
Affiliation:
Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, U.S.A. (email: [email protected])
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Abstract

It is shown here that given a discrete (and infinite) set of points in the plane, it is possible to arrange them on a polygonal path so that every angle on the polygonal path is at least 9∘. This has been known to hold for finite sets (with 20∘). The main result holds for discrete sets in higher dimensions as well, with a smaller bound on the angle.

MSC classification

Secondary: 52C10: Erdős problems and related topics of discrete geometry 52C30: Planar arrangements of lines and pseudolines
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 26 - 34
Copyright
Copyright © University College London 2010

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References

[1]Bárány, I., Pór, A. and Valtr, P., Paths without small angles. SIAM J. Discrete Math. (2009), to appear.Google Scholar
[2]Dumitrescu, A., 2005, personal communication.Google Scholar
[3]Fekete, S., Geometry and the traveling salesman problem, PhD Thesis, Department of Combinatorics and Optimization, The University of Waterloo, 1992.Google Scholar
[4]Fekete, S. and Woeginger, G. J., Angle-restricted tours in the plane. Comput. Geom. 8 (1997), 195–218.Google Scholar
[5]Kynčl, J., 2009, personal communication.Google Scholar