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

Paper

Abstract

Erdos, Herzog and Piranian asked whether, for

points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular

-gon maximizes their product of all pairs of distances. Recently, it was discovered that, for every even

, a regular

-gon is not a maximizer. However, the discovered improvement turns out to be very small. Indeed, for a fixed diameter of

, let

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

-gon,

for even

. The discovered arrangements have proven

thus far, and it was not known whether one can have

for some

and all sufficiently large even

. In this note, we show that indeed

for even

which settles this conjecture. Other arrangements with higher conjectured

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

are known and we do not know whether they exist.