On halving line arrangements Original Research Article

Given a set of n points in general position in the plane, where n is even, a halving line is a line going through two of the points and cutting the remaining set of n−2 points in half. Let h(n) denote the maximum number of halving lines that can be realized by a planar set of n points. The problem naturally generalizes to pseudoconfigurations; denote the maximum number of halving pseudolines over all pseudoconfigurations of size n by ĥ(n). We prove that ĥ(12)=18 and that the pseudoconfiguration on 12 points with the largest number of halving pseudolines is unique up to isomorphism; this pseudoconfiguration is realizable, implying h(12)=18. We show several structural results that substantially reduce the computational effort needed to obtain the exact value of ĥ(n) for larger n. Using these techniques, we enumerate all topologically distinct, simple arrangements of 10 pseudolines with a marked cell. We also prove that h(14)=22 using certain properties of degree sequences of halving edges graphs.