2025/12/16 by Sothanaphan, Nat
#52C10 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2512.14251
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 ≥ 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 Δ 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, Δ= nn for even n. The discovered arrangements have proven Δ= (1+o(1))nn thus far, and it was not known whether one can have Δ≥ C nn for some C > 1 and all sufficiently large even n. In this note, we show that indeed \liminfn→∞ Δmax/nn > 1.037 for even n which settles this conjecture. Other arrangements with higher conjectured Δ/nn 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 Δ/nn → ∞ are known and we do not know whether they exist.