Universal sequences of lines in ℝd

One of the most important and useful examples in discrete geometry is a finite sequence of points on the moment curve γ ( t ) = ( t, t 2 , t 3 , …, t d ) or, more generally, on a strictly monotone curve in ℝ d . These sequences as well as the ambient curve itself can be described in terms of 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. 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 of 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 continuous example of universal family of lines. The case d = 4 is left as an open question.