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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Acknowledgements
The research of I.B. was partially supported by Hungarian National Research Grants 131529, 131696 and 133819, and the research of G.K. by ERC advanced grant 834735 and by ISF grant 2669/21. We thank the anonymous referee for careful reading and for useful comments.
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Dedicated to Nati Linial for his friendship and influence, and for his vision of graphs as geometric objects
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Bárány, I., Kalai, G. & Pór, A. Universal sequences of lines in ℝd. Isr. J. Math. 256, 35–60 (2023). https://doi.org/10.1007/s11856-023-2504-x
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DOI: https://doi.org/10.1007/s11856-023-2504-x