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

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.

21 pages, 2 figures