{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/roudneff-s-conjecture-in-dimension-4","title":"Roudneff's Conjecture in Dimension $4$","arxiv_id":"2303.14212","date":"2023-03-24","proceeding":null,"authors":["Rangel Hernández-Ortiz","Kolja Knauer","Luis Pedro Montejano","Manfred Scheucher"],"abstract":"J.-P. Roudneff conjectured in 1991 that every arrangement of $n \\ge 2d+1\\ge 5$ pseudohyperplanes in the real projective space $\\mathbb{P}^d$ has at most $\\sum_{i=0}^{d-2} \\binom{n-1}{i}$ complete cells (i.e., cells bounded by each hyperplane). The conjecture is true for $d=2,3$ and for arrangements arising from Lawrence oriented matroids. The main result of this manuscript is to show the validity of Roudneff's conjecture for $d=4$. Moreover, based on computational data we conjecture that the maximum number of complete cells is only obtained by cyclic arrangements.","url_abs":"https://arxiv.org/abs/2303.14212v1","url_pdf":"https://arxiv.org/pdf/2303.14212v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"roudneff-s-conjecture-in-dimension-4","repo_url":"https://github.com/manfredscheucher/supplemental-roudneff4","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}