An improved lower bound to Erdos' problem concerning products of distances for fixed diameter

TL;DR

This paper establishes a new lower bound greater than 1.037 for the maximum product of all pairs of distances among points with fixed diameter, advancing understanding of Erdős' problem and disproving the optimality of regular polygons.

Contribution

It proves that the maximum product of distances exceeds the regular polygon value by a factor greater than 1.037 for large even n, providing a significant improvement in lower bounds.

Findings

  

The lower bound for the maximum product of distances is greater than 1.037 times n^n for large even n.

Regular n-gons are not optimal for maximizing the product of all pairwise distances.

Abstract

Erdos, Herzog and Piranian asked whether, for $n$ points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular $n$-gon maximizes their product of all pairs of distances. Recently, it was discovered that, for every even $n \geq 4$, a regular $n$-gon is not a maximizer. However, the discovered improvement turns out to be very small. Indeed, for a fixed diameter of $2$, let $\Delta$ be the square of the product of all pairs of distances (the "square" is here due to connections with polynomial discriminants). Then, for a regular $n$-gon, $\Delta = n^n$ for even $n$. The discovered arrangements have proven $\Delta = (1+o(1))n^n$ thus far, and it was not known whether one can have $\Delta \geq C n^n$ for some $C > 1$ and all sufficiently large even $n$. In this note, we show that indeed $\liminf_{n\to\infty} \Delta_{\max}/n^n > 1.037$ for even $n$ which settles this conjecture. Other arrangements with higher conjectured $\Delta/n^n$ values are in fact known, but we have not been able to obtain proofs that they have large products of distances. Finally, no arrangements such that $\Delta/n^n \to \infty$ are known and we do not know whether they exist.