An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of Points
In this paper, we give examples that improve the lower bound on the maximum number of halving lines for sets in the plane with 35, 59, 95, and 97 points and, as a consequence, we improve the current best upper bound of the rectilinear crossing number for sets in the plane with 35, 59, 95, and 97 poi...
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MDPI AG
2025-01-01
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| author | Javier Rodrigo Mariló López Danilo Magistrali Estrella Alonso |
| author_facet | Javier Rodrigo Mariló López Danilo Magistrali Estrella Alonso |
| author_sort | Javier Rodrigo |
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| description | In this paper, we give examples that improve the lower bound on the maximum number of halving lines for sets in the plane with 35, 59, 95, and 97 points and, as a consequence, we improve the current best upper bound of the rectilinear crossing number for sets in the plane with 35, 59, 95, and 97 points, provided that a conjecture included in the literature is true. As another consequence, we also improve the lower bound on the maximum number of halving pseudolines for sets in the plane with 35 points. These examples, and the recursive bounds for the maximum number of halving lines for sets with an odd number of points achieved, give a new insight in the study of the rectilinear crossing number problem, one of the most challenging tasks in Discrete Geometry. With respect to this problem, it is conjectured that, for all <i>n</i> multiples of 3, there are 3-symmetric sets of <i>n</i> points for which the rectilinear crossing number is attained. |
| format | Article |
| id | doaj-art-d13bd0f37e2c4189b0c3f67c9ccf9ca3 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-d13bd0f37e2c4189b0c3f67c9ccf9ca32025-01-24T13:22:18ZengMDPI AGAxioms2075-16802025-01-011416210.3390/axioms14010062An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of PointsJavier Rodrigo0Mariló López1Danilo Magistrali2Estrella Alonso3Departamento de Matemática Aplicada, E.T.S. de Ingeniería, Universidad Pontificia Comillas de Madrid, 28015 Madrid, SpainDepartamento de Matemática e Informática Aplicadas a las Ingenierías Civil y Naval de la E.T.S.I. Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040 Madrid, SpainDepartamento de Matemática Aplicada, E.T.S. de Ingeniería, Universidad Pontificia Comillas de Madrid, 28015 Madrid, SpainDepartamento de Matemática Aplicada, E.T.S. de Ingeniería, Universidad Pontificia Comillas de Madrid, 28015 Madrid, SpainIn this paper, we give examples that improve the lower bound on the maximum number of halving lines for sets in the plane with 35, 59, 95, and 97 points and, as a consequence, we improve the current best upper bound of the rectilinear crossing number for sets in the plane with 35, 59, 95, and 97 points, provided that a conjecture included in the literature is true. As another consequence, we also improve the lower bound on the maximum number of halving pseudolines for sets in the plane with 35 points. These examples, and the recursive bounds for the maximum number of halving lines for sets with an odd number of points achieved, give a new insight in the study of the rectilinear crossing number problem, one of the most challenging tasks in Discrete Geometry. With respect to this problem, it is conjectured that, for all <i>n</i> multiples of 3, there are 3-symmetric sets of <i>n</i> points for which the rectilinear crossing number is attained.https://www.mdpi.com/2075-1680/14/1/62discrete geometryhalving linesrectilinear crossing numberoptimization |
| spellingShingle | Javier Rodrigo Mariló López Danilo Magistrali Estrella Alonso An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of Points Axioms discrete geometry halving lines rectilinear crossing number optimization |
| title | An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of Points |
| title_full | An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of Points |
| title_fullStr | An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of Points |
| title_full_unstemmed | An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of Points |
| title_short | An Improvement of the Lower Bound on the Maximum Number of Halving Lines for Sets in the Plane with an Odd Number of Points |
| title_sort | improvement of the lower bound on the maximum number of halving lines for sets in the plane with an odd number of points |
| topic | discrete geometry halving lines rectilinear crossing number optimization |
| url | https://www.mdpi.com/2075-1680/14/1/62 |
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