Vol. 8, No. 2, 2015

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ISSN: 1944-4184 (e-only)
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Crossings of complex line segments

Samuli Leppänen

Vol. 8 (2015), No. 2, 285–294
Abstract

The crossing lemma holds in 2 because a real line separates the plane into two disjoint regions. In 2 removing a complex line keeps the remaining point-set connected. We investigate the crossing structure of affine line segment-like objects in 2 by defining two notions of line segments between two points and give computational results on combinatorics of crossings of line segments induced by a set of points. One way we define the line segments motivates a related problem in 3, which we introduce and solve.

Keywords
discrete geometry, crossing inequality
Mathematical Subject Classification 2010
Primary: 51M05, 51M30, 52C35
Secondary: 51M04
Milestones
Received: 16 July 2013
Revised: 22 February 2014
Accepted: 23 February 2014
Published: 3 March 2015

Communicated by Kenneth S. Berenhaut
Authors
Samuli Leppänen
Department of Mathematics
University of British Columbia
Vancouver BC V6T 1Z2
Canada