{"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/crescent-configurations-in-normed-spaces","title":"Crescent configurations in normed spaces","arxiv_id":"1909.08769","date":"2019-09-19","proceeding":null,"authors":["Sara Fish","Dylan King","Steven J. Miller","Eyvindur A. Palsson","Catherine Wahlenmayer"],"abstract":"We study the problem of crescent configurations, posed by Erd\\H{o}s in 1989. A crescent configuration is a set of $n$ points in the plane such that: 1) no three points lie on a common line, 2) no four points lie on a common circle, 3) for each $1 \\leq i \\leq n - 1$, there exists a distance which occurs exactly $i$ times. Constructions of sizes $n \\leq 8$ have been provided by Liu, Pal\\'{a}sti, and Pomerance. Erd\\H{o}s conjectured that there exists some $N$ for which there do not exist crescent configurations of size $n$ for all $n \\geq N$. We extend the problem of crescent configurations to general normed spaces $(\\mathbb{R}^2, \\| \\cdot \\|)$ by studying strong crescent configurations in $\\| \\cdot \\|$. In an arbitrary norm $\\|\\cdot \\|$, we construct a strong crescent configuration of size 4. We also construct larger strong crescent configurations in the Euclidean, taxicab, and Chebyshev norms, of sizes $n \\leq 6$, $n \\leq 8$, and $n \\leq 8$ respectively. When defining strong crescent configurations, we introduce the notion of line-like configurations in $\\|\\cdot \\|$. A line-like configuration in $\\|\\cdot \\|$ is a set of points whose distance graph is isomorphic to the distance graph of equally spaced points on a line. In a broad class of norms, we construct line-like configurations of arbitrary size. Our main result is a crescent-type result about line-like configurations in the Chebyshev norm. A line-like crescent configuration is a line-like configuration for which no three points lie on a common line and no four points lie on a common $\\|\\cdot \\|$ circle. We prove that for $n \\geq 7$, every line-like crescent configuration of size $n$ in the Chebyshev norm must have a rigid structure. Specifically, it must be a perpendicular perturbation of equally spaced points on a horizontal or vertical line.","url_abs":"https://arxiv.org/abs/1909.08769v2","url_pdf":"https://arxiv.org/pdf/1909.08769v2.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":"crescent-configurations-in-normed-spaces","repo_url":"https://github.com/the-set-of-sets/l1_linfty","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"crescent-configurations-in-normed-spaces","repo_url":"https://github.com/the-set-of-sets/nin","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}