{"paper":{"title":"Concave cocirculations in a triangular grid","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander V. Karzanov","submitted_at":"2003-04-21T12:16:01Z","abstract_excerpt":"Let $G=(V(G),E(G))$ be a planar digraph embedded in the plane in which all inner faces are equilateral triangles (with three edges in each), and let the union $\\Rscr$ of these faces forms a convex polygon. The question is: given a function $\\sigma$ on the boundary edges of $G$, does there exist a concave function $f$ on $\\Rscr$ which is affinely linear within each bounded face and satisfies $f(v)-f(u)=\\sigma(e)$ for each boundary edge $e=(u,v)$?\n  The functions $\\sigma$ admitting such an $f$ form a polyhedral cone $C$, and when the region $\\Rscr$ is a triangle, $C$ turns out to be exactly the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0304289","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}