On the maximum product of distances of diameter $2$ point sets
Combinatorics
2026-03-10 v1 Complex Variables
Metric Geometry
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 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 -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.
Keywords
Cite
@article{arxiv.2603.07088,
title = {On the maximum product of distances of diameter $2$ point sets},
author = {Stijn Cambie and Arne Decadt and Yanni Dong and Tao Hu and Quanyu Tang},
journal= {arXiv preprint arXiv:2603.07088},
year = {2026}
}
Comments
43 pages (out of which 17 are appendix) 6+2 figures and 1 table