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

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.