Rederiving the Upper Bound for Halving Edges using Cardano's Formula

Abstract

In this paper we rederive an old upper bound on the number of halving edges present in the halving graph of an arbitrary set of $n$ points in 2-dimensions which are placed in general position. We provide a different analysis of an identity discovered by Andrejak et al, to rederive this upper bound of $O(n^{4/3})$. In the original paper of Andrejak et al. the proof is based on a naive analysis whereas in this paper we obtain the same upper bound by tightening the analysis thereby opening a new door to derive these upper bounds using the identity. Our analysis is based on a result of Cardano for finding the roots of a cubic equation. We believe that our technique has the potential to derive improved bounds on the number of halving edges.