# Some remarks on the midrange crossing constant

## TL;DR

This paper verifies an upper bound on the midrange crossing constant originally proposed by Pach and Toth, using a different method based on Moon's result, and questions whether this bound is tight.

## Contribution

It provides a new verification of the upper bound on the midrange crossing constant using Moon's result, differing from previous methods.

## Key findings

- Confirmed the upper bound of 8/(9π^2) on the midrange crossing constant.
- Established a new proof technique based on Moon's result.
- Raised the open question about the tightness of the bound.

## Abstract

We verify an upper bound of Pach and T\'oth [Combinatorica 17(1997), 427-439, Discrete and Computational Geometry 36(2006), 527-552] on the midrange crossing constant. Details of their $\frac{8}{9\pi^2}$ upper bound have not been available. Our verification is different from their method and hinges on a result of Moon [J. Soc. Indust. Appl. Math. 13(1965), 506-510]. As Moon's result is optimal, we raise the question whether the midrange crossing constant is $\frac{8}{9\pi^2}$.