Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Upper bounds for centerlines   In 2008, Bukh, Matousek, 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 (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 R^d for which
no line in R^d lies at depth larger 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 R^3.

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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg