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

TL;DR

This paper investigates the maximum product of distances in point sets with diameter 2, proving convexity sufficiency, analyzing the structure of extremal configurations, and proposing improved constructions over regular polygons.

Contribution

It demonstrates that extremal point sets can be assumed convex and provides new constructions and structural insights into the extremal configurations.

Findings

Convex polygons suffice for extremal configurations.

Regular n-gons are not always optimal, and improved constructions exist.

Structural properties of the diameter graph are characterized.

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.