Empty pentagons in point sets with collinearities
Combinatorics
2015-02-17 v1 Computational Geometry
Discrete Mathematics
Abstract
An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.
Cite
@article{arxiv.1207.3633,
title = {Empty pentagons in point sets with collinearities},
author = {János Barát and Vida Dujmović and Gwenaël Joret and Michael S. Payne and Ludmila Scharf and Daria Schymura and Pavel Valtr and David R. Wood},
journal= {arXiv preprint arXiv:1207.3633},
year = {2015}
}
Comments
15 pages, 11 figures