{"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/voronoi-diagrams-in-the-hilbert-metric","title":"Voronoi Diagrams in the Hilbert Metric","arxiv_id":"2112.03056","date":"2021-12-06","proceeding":null,"authors":["Auguste H. Gezalyan","David M. Mount"],"abstract":"The Hilbert metric is a distance function defined for points lying within a convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to any convex set, and it has numerous applications in the analysis and processing of convex bodies. In this paper, we study the geometric and combinatorial properties of the Voronoi diagram of a set of point sites under the Hilbert metric. Given any convex polygon $K$ bounded by $m$ sides, we present two algorithms (one randomized and one deterministic) for computing the Voronoi diagram of an $n$-element point set in the Hilbert metric induced by $K$. Our randomized algorithm runs in $O(m n + n (\\log n)(\\log m n))$ expected time, and our deterministic algorithm runs in time $O(m n \\log n)$. Both algorithms use $O(m n)$ space. We show that the worst-case combinatorial complexity of the Voronoi diagram is $\\Theta(m n)$.","url_abs":"https://arxiv.org/abs/2112.03056v1","url_pdf":"https://arxiv.org/pdf/2112.03056v1.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":"voronoi-diagrams-in-the-hilbert-metric","repo_url":"https://github.com/caesardai/voronoi_in_hilbert","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}