{"id":"palomar/PALOMAR-2026-10-27-000022","url":"https://github.com/pragyaangaur/Polyhex-Hull/tree/aad7fdec228045c16c0e5bd0794aff692a5d527a","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-27-000022-v1.json","repo":"https://github.com/pragyaangaur/Polyhex-Hull","tags":["math.CO","math.MG","msc:05B50","msc:52A38","msc:52C20"],"score":0,"title":"Kurz's conjecture on the maximum area of the convex hull of a polyhex","trust":"high","domain":"math.CO","source":"palomar","authors":["Pragyaan Gaur"],"reviews":0,"subject":"math.CO","summary":"A polyhex is an edge-to-edge connected system of cells of the regular hexagonal tiling. Kurz (Convex hulls of polyominoes, Beiträge Algebra Geom. 49 (2008), Conjecture 2) conjectured that the convex hull of a polyhex with n cells of unit area has area at most ⌊n² + 14n/3 + 1⌋ / 6. This formalization proves the conjecture, and the sharper bound ⌈n² + 14n/3⌉ / 6, which differs from Kurz's bound by 1/6 when 3 divides n and agrees with it otherwise. The statements are made in the Euclidean plane for the regular hexagonal tiling, with area measured in units of one cell. The proof writes the hull…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-27","reactions":{},"formalized":true,"formalizes":[{"title":"The maximum area of the convex hull of a polyhex","identifier":"arXiv:2609.30310","relationship":"formalizes"}]}