Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10403067" target="_blank" >RIV/00216208:11320/19:10403067 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=MOUjoxqJui" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=MOUjoxqJui</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00454-018-9970-7" target="_blank" >10.1007/s00454-018-9970-7</a>

Result language

angličtina

Original language name

Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences

Original language description

Let d and k be integers with 1kd-1. Let 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 K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional nxxn grid is at least (nd(d-k)/(d-1)-epsilon) and at most O(nd(d-k)/(d-1)), where epsilon&gt;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 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 d3 and epsilon(0,1), we show that there is an integer r=r(d,epsilon) such that for all positive integers n,m the following statement is true. There is a set of n points in Rd and an arrangement of m hyperplanes in Rd with no Kr,r in their incidence graph and with at least ((mn)1-(2d+3)/((d+2)(d+3))-epsilon) incidences if d is odd and ((mn)1-(2d2+d-2)/((d+2)(d2+2d-2))-epsilon) incidences if d is even.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GA14-14179S" target="_blank" >GA14-14179S: Algorithmic, structural and complexity aspects of configurations in the plane</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2019

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Discrete and Computational Geometry

ISSN

0179-5376

e-ISSN

—

Volume of the periodical

61

Issue of the periodical within the volume

2

Country of publishing house

US - UNITED STATES

Number of pages

30

Pages from-to

325-354

UT code for WoS article

000456720300006

EID of the result in the Scopus database

2-s2.0-85041829923