Everywhere unbalanced configurations

An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number \(k\) such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most \(k\). We give a n...

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Bibliographic Details
Published in:arXiv.org 2023-08
Main Authors: Conlon, David, Jeck Lim
Format: Article
Language:English
Online Access:Get full text
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Summary:An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number \(k\) such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most \(k\). We give a negative answer to a natural variant of this problem, showing that for every natural number \(k\) there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least \(k\).
ISSN:2331-8422