A Bichromatic Incidence Bound and an Application

Details A Bichromatic Incidence Bound and an Application A Bichromatic Incidence Bound and an Application

2010-06-19

arxiv.org/abs/1006.3878v6

Discrete & Computational Geometry, Volume 46, Number 4, 2011, 611-625

Mathematics

Combinatorics

12 pages

Scientific paper

10.1007/s00454-011-9367-3

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.

Affiliated with

Also associated with

No associations

Rating

A Bichromatic Incidence Bound and an Application does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Bichromatic Incidence Bound and an Application, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Bichromatic Incidence Bound and an Application will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-431654