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\({\varvec{k}}\)-Bisectors of Finite Planar Sets

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Abstract

Let V be a finite set of points in the plane, not all on one line, and let l be a line that contains at least 2 points of V. We say that l is a k -bisector of V if there are at least k points of V on each one of the two open half-planes bounded by l. For \(k \ge 6\) we construct planar sets of \(2k+4\) points having no k-bisector (this might be best possible). Furthermore, we show that if \(|V| > 3 k\), then in every triangulation of convV with vertex set V there is an edge whose loading line is a k-bisector of V. This is best possible for all k.

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Notes

  1. Thus \(\varphi (6) = 16\), a non-trivial byproduct.

  2. Note that our “k” is equivalent to \(3k-1\) in l.c., and that the example of \(n=6k+1\) points without a (\(3k-1\))-bisector given there satisfies, in terms of our “k”, \(n=6k+1=2(3k-1)+3=2 \cdot ``k\)”\(+3\).

  3. Using again Helly’s Theorem, it is shown there that the value of the function analogous to \(\varphi \) in 3-space, with “k-bisector line” replaced by “k-bisector plane”, is exactly 4k; so the problem is much simpler in 3-space than in the plane.

  4. Claim 3.1 remains true when the condition “\(p \in V {\setminus } \cup \mathcal{L}(V)\)” is replaced by the weaker condition “\(p \in V {\setminus } \mathrm{conv}_2 V\)”, where conv\(_2 V\) is the union of all segments with endpoints in V. We do not need this refined result.

References

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University, Jerusalem, Israel

    Yaakov S. Kupitz & Micha A. Perles

  2. Mathematics Faculty, University of Technology, 09107, Chemnitz, Germany

    Horst Martini

Authors
  1. Yaakov S. Kupitz
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  2. Horst Martini
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  3. Micha A. Perles
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Correspondence to Horst Martini.

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Kupitz, Y.S., Martini, H. & Perles, M.A. \({\varvec{k}}\)-Bisectors of Finite Planar Sets. Graphs and Combinatorics 33, 981–990 (2017). https://doi.org/10.1007/s00373-017-1799-y

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  • DOI: https://doi.org/10.1007/s00373-017-1799-y

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