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

TL;DR

This paper rederives the $O(n^{4/3})$ upper bound on halving edges in 2D point sets by applying Cardano's formula to analyze an existing identity, offering a tighter and potentially more powerful approach.

Contribution

It introduces a novel analysis method using Cardano's formula to tighten the upper bound on halving edges, improving upon previous naive proofs.

Findings

Rederived the $O(n^{4/3})$ upper bound using a new analysis.

Applied Cardano's formula to analyze the roots of a cubic related to halving edges.

The technique opens possibilities for deriving improved bounds in geometric graph theory.

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.