On the maximum product of distances of diameter $2$ point sets

Abstract

We consider a problem posed by Erd\H{o}s, Herzog and Piranian on the maximum product of distances of a point set of order $n$ with a given diameter. We prove that it is sufficient to consider convex polygons and obtain results on the structure of the diameter graph. We also give constructions that drastically improve on the regular $n$-gons, sketching what the extremal polygons should look like, while presenting results indicating that one cannot hope to characterize the extremal polygons in general for even orders.