Permutations of point sets in $\mathbb{R}^d$

TL;DR

This paper investigates the maximum and minimum number of orderings of point sets in various dimensions induced by distance measurements from vantage points, revealing configurations that achieve these extremal values and exploring related geometric and application aspects.

Contribution

It extends known results on orderings induced by vantage points to new configurations, dimensions, and spherical cases, and explores extremal and intermediate cases.

Findings

Maximum number of orderings equals sum of unsigned Stirling numbers of the first kind.

Minimum value of orderings is 2n-2, achieved by equally spaced points on a line.

Connections made between spherical and planar configurations, with open problems proposed.

Abstract

Given a set $S$ consisting of $n$ points in $\mathbb{R}^d$ and one or two vantage points, we study the number of orderings of $S$ induced by measuring the distance (for one vantage point) or the average distance (for two vantage points) from the vantage point(s) to the points of $S$ as the vantage points move through $\mathbb{R}^d.$ With one vantage point, a theorem of Good and Tideman \cite{MR505547} shows the maximum number of orderings is a sum of unsigned Stirling numbers of the first kind. We show that the minimum value in all dimensions is $2n-2,$ achieved by $n$ equally spaced points on a line. We investigate special configurations that achieve intermediate numbers of orderings in the one--dimensional and two--dimensional cases. We also treat the case when the points are on the sphere $S^2,$ connecting spherical and planar configurations. We briefly consider an application using weights suggested by an application to social choice theory. We conclude with several open problems that we believe deserve further study.