A Bichromatic Incidence Bound and an Application
Abstract
We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = \Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal and Aronov. We use this incidence bound to prove that a set of n points, no more than n-k of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy's on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.
Keywords
Cite
@article{arxiv.1006.3878,
title = {A Bichromatic Incidence Bound and an Application},
author = {Ben D. Lund and George B. Purdy and Justin W. Smith},
journal= {arXiv preprint arXiv:1006.3878},
year = {2012}
}
Comments
12 pages