## Abstract

Let d and k be integers with 1 ≤ k≤ d- 1. Let Λ 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 Λ ∩ K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n× ⋯ × n grid is at least Ω (n^{d} ^{(} ^{d} ^{-} ^{k} ^{)} ^{/} ^{(} ^{d} ^{-} ^{1} ^{)} ^{-} ^{ε}) and at most O(n^{d} ^{(} ^{d} ^{-} ^{k} ^{)} ^{/} ^{(} ^{d} ^{-} ^{1} ^{)}) , where ε> 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book by Brass et al. (Research problems in discrete geometry, Springer, New York, 2005). We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover Λ ∩ K. We use these new results to improve the best known lower bound for the maximum number of point–hyperplane incidences by Brass and Knauer (Comput Geom 25(1–2):13–20, 2003). For d≥ 3 and ε∈ (0 , 1) , we show that there is an integer r= r(d, ε) such that for all positive integers n, m the following statement is true. There is a set of n points in R^{d} and an arrangement of m hyperplanes in R^{d} with no K_{r} _{,} _{r} in their incidence graph and with at least Ω ((mn) ^{1} ^{-} ^{(} ^{2} ^{d} ^{+} ^{3} ^{)} ^{/} ^{(} ^{(} ^{d} ^{+} ^{2} ^{)} ^{(} ^{d} ^{+} ^{3} ^{)} ^{)} ^{-} ^{ε}) incidences if d is odd and Ω((mn)1-(2d2+d-2)/((d+2)(d2+2d-2))-ε) incidences if d is even.

Original language | English |
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Pages (from-to) | 325-354 |

Number of pages | 30 |

Journal | Discrete and Computational Geometry |

Volume | 61 |

Issue number | 2 |

DOIs | |

State | Published - 15 Mar 2019 |

Externally published | Yes |

## Keywords

- Covering
- Lattice point
- Linear subspace
- Point–hyperplane incidence

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics