Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 45, No. 1, pp. 133-153 (2004)

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Unbounded regions in an arrangement of lines in the plane

Alan West

School of Mathematics, University of Leeds, Leeds LS2 9JT, England

Abstract: We take a set $\Omega$ of $n$ points and an arrangement $\Sigma$ of $m$ lines in the plane which avoid these points but separate any two of them. We suppose these satisfy the following unboundedness property: for each point $x \in \Omega$ there is a homotopy from $\Sigma$ to ${\Sigma}'$ avoiding $\Omega$ so that $x$ is in an unbounded component of the complement of ${\Sigma}'$. It is proved that then $n \leq 2m$. This result is required to partially solve a problem in differential geometry which is described briefly.

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Electronic version published on: 5 Mar 2004. This page was last modified: 4 May 2006.

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