Barany, Imre; Kalai, Gil; Por, Attila; (2023) Universal sequences of lines in ℝd. Israel Journal of Mathematics , 256 (1) pp. 35-60. 10.1007/s11856-023-2504-x.

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Abstract

One of the most important and useful examples in discrete geometry is a finite sequence of points on the moment curve γ(t) = (t, t2, t3, …, td) or, more generally, on a strictly monotone curve in ℝd. 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 2d−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.

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