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Covering lattice points by subspaces and counting point-hyperplane incidences

Banff International Research Station for Mathematical Innovation and Discovery

Let d and k be integers with 1 <= k <= d-1. Let Lambda be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in the intersection of Lambda with K. In particular, our results imply that the minimum number of kdimensional linear subspaces needed to cover the d-dimensional n * ... * n$ grid is at least Omega(n^(d(d-k)/(d-1)-epsilon)) and at most O(n^(d(d-k)/(d-1))), where epsilon > 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach. We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer. This is a joint work with Josef Cibulka and Pavel Valtr.

Mathematics; Convex and discrete geometry; Combinatorics

video/mp4

Author affiliation: Ben Gurion University of the Negev

2018-08-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/66701

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/