The Maximum Number of Halving Lines for Finite Planar Sets—New Lower Bounds

Abstract

In this paper, we deal with a challenging problem in Discrete Geometry. The problem is to maximize the number of halving lines for finite sets in the plane. With respect to this, we improve the lower bounds for the maximum number of halving lines for sets in the plane of n points (hn), for some values of n with n ≤ 100, expanding the range of the values of n for which the authors achieved an improvement in preliminary works. To do this, we strengthen previous results of the authors about recursive bounds concerning hn. These improvements are important due to the relation of the number of halving lines for finite sets with another classical problem in Discrete Geometry, the rectilinear crossing number problem.