Crossings of complex line segments

Leppänen, Samuli

doi:10.2140/involve.2015.8.285

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Crossings of complex line segments

Samuli Leppänen

Vol. 8 (2015), No. 2, 285–294

DOI: 10.2140/involve.2015.8.285

Abstract

The crossing lemma holds in $ℝ^{2}$ because a real line separates the plane into two disjoint regions. In $ℂ^{2}$ removing a complex line keeps the remaining point-set connected. We investigate the crossing structure of affine line segment-like objects in $ℂ^{2}$ 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 $ℝ^{3}$ , 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

Vol. 8, No. 2, 2015