Optimal point layouts maximize distance products with fixed sum
Optimal Configurations for Mordell's Problem
Symbolic Computation
Summary
This paper studies how to place points on a plane so that the product of the squared distances between every pair of points is as large as possible, given a fixed total distance from the origin. For small numbers of points, it's known that placing them evenly around a circle works best. The authors prove that for six to nine points, the best layout is putting one point at the center and the others evenly around a circle. They also rule out previous guesses for larger numbers of points. This work connects to earlier problems about minimizing the space taken by points.
What this means in practice
- •For geometric algorithm designers: Design algorithms that arrange points optimally under distance constraints using proven maximal configurations for sets of six to nine points.
- •For signal processing engineers: Utilize optimal point configurations to improve designs relying on complex-plane point placements for antenna arrays or phase encoding schemes.
A theory result. No direct application yet.
Authors
Xiang Ji, Tiancheng Xu, Wensheng Yu
Abstract
Mordell's problem asks for the largest product of squared pairwise distances among $n$ points in the complex plane with a prescribed sum of squared distances from the origin. When this sum equals $n$, the maximum for $2\le n\le5$ is known to be $n^n$, attained precisely by the $n$th roots of unity up to rotation and relabeling. For $n\ge6$, Chalk showed that a regular polygon together with its center gives a larger product than the regular $n$-gon. We prove that for $6\le n\le9$ the maximum is $(n-1)^{n-1}r_n^{n(n-1)}$, where $r_n=\sqrt{n/(n-1)}$. Up to the same symmetries, the unique maximizer consists of the origin and the vertices of a regular $(n-1)$-gon of radius $r_n$. For six points, algebraic elimination reduces the proof to an exact comparison of stationary values. For seven points, we compare all twelve classes in a published classification of equilibria. For eight and nine points, rigorous interval arithmetic establishes an inequality for every admissible pair. Summing over pairs gives the sharp global bound, and the equality conditions determine the configuration. An explicit ten-point comparison and a known instability result together exclude Chalk's configuration for every $n\ge10$. For each $n$ from two through eight, Mordell's problem and the minimum-diameter problem studied by Bateman and Erdős have a common optimal configuration.