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Peer-reviewedCC BY 4.0Axioms

The Maximum Number of Halving Lines for Finite Planar Sets—New Lower Bounds

  • Javier Rodrigo1
  • Sagrario Lantarón2
  • María Dolores López2
  1. 1Department of Applied Mathematics, Technical School of Engineering, Comillas Pontifical University, Calle de Alberto Aguilera 25, 28015 Madrid, Spain
  2. 2Department of Mathematics and Computer Science Applied to Civil and Naval Engineering, Universidad Politécnica de Madrid, Calle de Profesor Aranguren 3, 28040 Madrid, Spain
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Abstract

In this paper, we deal with a challenging problem in Discrete Geometry. The problem is to maximize the number of halving lines for finite sets in the plane. With respect to this, we improve the lower bounds for the maximum number of halving lines for sets in the plane of n points (hn), for some values of n with n ≤ 100, expanding the range of the values of n for which the authors achieved an improvement in preliminary works. To do this, we strengthen previous results of the authors about recursive bounds concerning hn. These improvements are important due to the relation of the number of halving lines for finite sets with another classical problem in Discrete Geometry, the rectilinear crossing number problem.

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Cite

BibTeXRIS
BibTeX
@article{rodrigo2026maximum,
  title = {{The Maximum Number of Halving Lines for Finite Planar Sets—New Lower Bounds}},
  author = {Javier Rodrigo and Sagrario Lantarón and María Dolores López},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100771},
  url = {https://www.mdpi.com/2075-1680/15/10/771},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - The Maximum Number of Halving Lines for Finite Planar Sets—New Lower Bounds
AU  - Javier Rodrigo
AU  - Sagrario Lantarón
AU  - María Dolores López
PY  - 2026
DA  - 2026/10/15
JO  - Axioms
DO  - 10.3390/axioms15100771
UR  - https://www.mdpi.com/2075-1680/15/10/771
AB  - In this paper, we deal with a challenging problem in Discrete Geometry. The problem is to maximize the number of halving lines for finite sets in the plane. With respect to this, we improve the lower bounds for the maximum number of halving lines for sets in the plane of n points (hn), for some values of n with n ≤ 100, expanding the range of the values of n for which the authors achieved an improvement in preliminary works. To do this, we strengthen previous results of the authors about recursive bounds concerning hn. These improvements are important due to the relation of the number of halving lines for finite sets with another classical problem in Discrete Geometry, the rectilinear crossing number problem.
SN  - 2075-1680
ER  -