Universal sequences of lines in $\mathbb R^d$

TL;DR

This paper explores universal sequences of lines in d-dimensional space, revealing that such sequences are limited in number and mostly unique, with an open question remaining in four dimensions.

Contribution

It provides bounds on the number of universal line sequences in d-space and shows their near-uniqueness except in four dimensions.

Findings

Number of universal line sequences is at least 2^{d-1}-2 and at most (d-1)!

In all dimensions except 4, there is essentially a unique continuous universal family of lines

The case d=4 remains an open problem.

Abstract

One of the most important and useful examples in discrete geometry is a finite sequence of points on the moment curve $\gamma(t)=(t,t^2,t^3,\dots ,t^d)$ or, more generally, on a {\it strictly monotone curve} in $\mathbb R^d$. These sequences as well as the ambient curve itself can be described in terms of {\it universality properties} and we will study the question: "What is a universal sequence of oriented and unoriented lines in $d$-space'' We give partial answers to this question, and to the analogous one for $k$-flats. Given a large integer $n$, it turns out that, like the case of points the number of universal configurations is bounded by a function of $d$, but unlike the case for points, there are a large number of distinct universal finite sequences of lines. We show that their number is at least $2^{d-1}-2$ and at most $(d-1)!$. However, like for points, in all dimensions except $d=4$, there is essentially a unique {\em continuous} example of a universal family of lines. The case $d=4$ is left as an open question.