A Note on the Number of Regions in a Line Arrangement

TL;DR

This paper establishes a new lower bound on the number of regions formed by line arrangements in the real projective plane, particularly when no more than two-thirds of the lines intersect at a single point.

Contribution

It introduces a novel lower bound for the number of regions in line arrangements using Bojanowski's inequality, under specific intersection constraints.

Findings

New lower bound for regions: at least (1/6)n^2 when no more than 2/3 lines intersect at a point.

Applicable to arrangements with limited intersection multiplicity.

Enhances understanding of geometric combinatorics in line arrangements.

Abstract

For an arrangement of $n$ lines in the real projective plane, we denote by $f$ the number of regions into which the real projective plane is divided by the lines. Using Bojanowski's inequality, we establish a new lower bound for $f$. In particular, we show that if no more than $\frac{2}{3}n$ lines intersect at any point, then $f \ge \frac{1}{6}n^{2}$