{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T11:12:59Z","timestamp":1759921979276,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,16]],"date-time":"2025-01-16T00:00:00Z","timestamp":1736985600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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 n multiples of 3, there are 3-symmetric sets of n points for which the rectilinear crossing number is attained.<\/jats:p>","DOI":"10.3390\/axioms14010062","type":"journal-article","created":{"date-parts":[[2025,1,16]],"date-time":"2025-01-16T06:46:10Z","timestamp":1737009970000},"page":"62","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"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"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3590-8791","authenticated-orcid":false,"given":"Javier","family":"Rodrigo","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada, E.T.S. de Ingenier\u00eda, Universidad Pontificia Comillas de Madrid, 28015 Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9974-7918","authenticated-orcid":false,"given":"Maril\u00f3","family":"L\u00f3pez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica e Inform\u00e1tica Aplicadas a las Ingenier\u00edas Civil y Naval de la E.T.S.I. Caminos, Canales y Puertos, Universidad Polit\u00e9cnica de Madrid, 28040 Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6337-2184","authenticated-orcid":false,"given":"Danilo","family":"Magistrali","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada, E.T.S. de Ingenier\u00eda, Universidad Pontificia Comillas de Madrid, 28015 Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6030-4239","authenticated-orcid":false,"given":"Estrella","family":"Alonso","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada, E.T.S. de Ingenier\u00eda, Universidad Pontificia Comillas de Madrid, 28015 Madrid, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,16]]},"reference":[{"key":"ref_1","unstructured":"Srivastava, J.N. (1973). Dissection graphs of planar point sets. A Survey of Combinatorial Theory, North-Holland."},{"key":"ref_2","unstructured":"Eppstein, D. (1992). Set of Points with Many Halving Lines, Technical Report ICS-TR-92-86, Department of Information and Computer Science, University of California."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s004540010022","article-title":"Point sets with many k-sets","volume":"26","year":"2001","journal-title":"Discret. Comput. Geom."},{"key":"ref_4","unstructured":"Goodman, J.E., Pach, J., and Pollack, R. (2008). An improved, simple construction of many halving edges. Surveys on Discrete and Computational Geometry: Twenty Years Later, AMS. Contemporary Mathematics."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/PL00009354","article-title":"Improved bounds for planar k-sets and related problems","volume":"19","author":"Dey","year":"1998","journal-title":"Discret. Comput. Geom."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Alonso, E., L\u00f3pez, M., and Rodrigo, J. (2024). An Improvement of the Upper Bound for the Number of Halving Lines of Planar Sets. Symmetry, 16.","DOI":"10.3390\/sym16070936"},{"key":"ref_7","first-page":"1240","article-title":"3-symmetric and 3-decomposable geometric drawings of Kn\u00a0Discret","volume":"158","author":"Cetina","year":"2010","journal-title":"Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00454-007-1325-8","article-title":"New lower bounds for the number of (<=k)-edges and the rectilinear crossing number of Kn","volume":"38","author":"Aichholzer","year":"2007","journal-title":"Discret. Comput. Geom."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1007\/s00454-012-9403-y","article-title":"On (<=k)-edges, crossings, and halving lines of geometric drawings of Kn","volume":"48","author":"Cetina","year":"2012","journal-title":"Discret. Comput. Geom."},{"key":"ref_10","unstructured":"Aichholzer, O. (2023, November 01). On the Rectilinear Crossing Number. Available online: http:\/\/www.ist.tugraz.at\/staff\/aichholzer\/crossings.html."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.endm.2018.06.052","article-title":"An improvement of the lower bound on the maximum number of halving lines in planar sets with 32 points","volume":"68","author":"Rodrigo","year":"2018","journal-title":"Electron. Notes Discret. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.dam.2021.05.029","article-title":"New algorithms and bounds for halving pseudolines","volume":"319","author":"Bereg","year":"2022","journal-title":"Discret. Appl. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/62\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:29:52Z","timestamp":1759919392000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/62"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,16]]},"references-count":12,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010062"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010062","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,16]]}}}