{"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/on-the-cardinality-of-sets-in-bf-r-d-obeying","title":"On the cardinality of sets in ${\\bf R}^d$ obeying a slightly obtuse angle bound","arxiv_id":"2007.13871","date":"2020-07-27","proceeding":null,"authors":["Tongseok Lim","Robert J. McCann"],"abstract":"In this paper we explicitly estimate the number of points in a subset $A \\subset \\R^{d}$ as a function of the maximum angle $\\angle A$ that any three of these points form, provided $\\angle A < \\theta_d := \\arccos(-\\frac 1 {d}) \\in (\\pi/2,\\pi)$. We also show $\\angle A < \\theta_d$ ensures that $A$ coincides with the vertex set of a convex polytope. This study is motivated by a question of Paul Erd\\H{o}s and indirectly by a conjecture of L\\'aszl\\'o Fejes T\\'oth.","url_abs":"https://arxiv.org/abs/2007.13871v4","url_pdf":"https://arxiv.org/pdf/2007.13871v4.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":"on-the-cardinality-of-sets-in-bf-r-d-obeying","repo_url":"https://github.com/tlim0213/tlim0213.github.io","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}