Everywhere unbalanced configurations

TL;DR

The paper disproves a longstanding conjecture in discrete geometry by constructing point sets with pseudoline arrangements where the point distribution across any pseudoline is highly unbalanced, showing such configurations can be arbitrarily unbalanced.

Contribution

It introduces a construction demonstrating that for any natural number k, there exists a point set with a pseudoline arrangement where the point imbalance exceeds k, refuting the existence of a universal balancing line.

Findings

Constructed point sets with arbitrarily large imbalance on pseudolines.

Showed the minimal size of such configurations is doubly exponential in k.

Confirmed the size bound is optimal up to a constant factor.

Abstract

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 $k$. Moreover, we may find such a configuration with at most $2^{2^{ck}}$ points, which, by a result of Pinchasi, is best possible up to the value of the constant $c$.