10.4230/LIPICS.SOCG.2017.15
Basit, Abdul
Abdul
Basit
Dvir, Zeev
Zeev
Dvir
Saraf, Shubhangi
Shubhangi
Saraf
Wolf, Charles
Charles
Wolf
On the Number of Ordinary Lines Determined by Sets in Complex Space
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2017
Incidences
Combinatorial Geometry
Designs
Polynomial Method
Aronov, Boris
Boris
Aronov
Katz, Matthew J.
Matthew J.
Katz
2017
2017-06-20
2017-06-20
2017-06-20
en
urn:nbn:de:0030-drops-71883
10.4230/LIPIcs.SoCG.2017
978-3-95977-038-5
1868-8969
10.4230/LIPIcs.SoCG.2017
LIPIcs, Volume 77, SoCG 2017
33rd International Symposium on Computational Geometry (SoCG 2017)
2017
77
15
15:1
15:15
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Aronov, Boris
Boris
Aronov
Katz, Matthew J.
Matthew J.
Katz
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2017
77
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
15 pages
542725 bytes
application/pdf
Creative Commons Attribution 3.0 Unported license
info:eu-repo/semantics/openAccess
Kelly's theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.
LIPIcs, Vol. 77, 33rd International Symposium on Computational Geometry (SoCG 2017), pages 15:1-15:15