Upper bounds for centerlines

In 2008, Bukh, Matoušek, and Nivasch conjectured that for every n-point set S in ℝd and every k, 0 ≤ k ≤ d-1, there exists a k-flat f in ℝd (a "centerflat") that lies at "depth" (k + 1)n/(k + d + 1) - O(1) in S, in the sense that every halfspace that contains f contains at least that many points of S. This claim is true and tight for k = 0 (this is Rado's centerpoint theorem), as well as for k = d-1 (trivial). Bukh et al. showed the existence of a (d - 2)-flat at depth (d - 1)n/(2d - 1) - O(1) (the case k = d- 2). In this paper we concentrate on the case k = 1 (the case of "centerlines"), in which the conjectured value for the leading constant is 2/(d + 2). We prove that 2/(d + 2) is an upper bound for the leading constant. Specifically, we show that for every fixed d and every n there exists an n-point set in ℝd for which no line in ℝd lies at depth greater than 2n/(d + 2) + o(n). This point set is the "stretched grid"—a set which has been previously used by Bukh et al. for other related purposes. Hence, in particular, the conjecture is now settled for ℝ3 .ISSN:1920-180