Skip to main content

On the Number of Ordinary Lines Determined by Sets in Complex Space

Author(s): Basit, Abdul; Dvir, Zeev; Saraf, Shubhangi; Wolf, Charles

To refer to this page use:
Abstract: Kelly’s theorem states that a set of n points affinely spanning ℂ3 must determine at least one ordinary complex line (a line incident to 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 stronger bounds that take advantage of the existence of lines with four or 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.
Publication Date: 2019
Citation: Basit, Abdul, Zeev Dvir, Shubhangi Saraf, and Charles Wolf. "On the Number of Ordinary Lines Determined by Sets in Complex Space." Discrete & Computational Geometry 61, no. 4 (2019): pp. 778-808. doi:10.1007/s00454-018-0039-4
DOI: 10.1007/s00454-018-0039-4
ISSN: 0179-5376
EISSN: 1432-0444
Pages: 778 - 808
Type of Material: Journal Article
Journal/Proceeding Title: Discrete & Computational Geometry
Version: Author's manuscript

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.