### Upper bounds for centerlines

#### Abstract

In 2008, Bukh, Matoušek, and Nivasch conjectured that for every

*n*-point set*S*in ℝ^{d}and every*k*, 0 ≤ k ≤*d*-1, there exists a*k*-flat*f*in ℝ^{d}(acenterflat) 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*/(2*d*- 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 ℝ^{d} for which no line in ℝ^{d} lies at depth greater than 2*n*/(*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 ℝ^{3} .

#### Full Text:

PDFDOI: http://dx.doi.org/10.20382/jocg.v3i1a2

ISSN: 1920-180X