{"paper":{"title":"Notions of maximality for integral lattice-free polyhedra: the case of dimension three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.OC"],"primary_cat":"math.CO","authors_text":"Gennadiy Averkov, Jan Kr\\\"umpelmann, Stefan Weltge","submitted_at":"2015-09-17T10:41:28Z","abstract_excerpt":"Lattice-free sets (convex subsets of $\\mathbb{R}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral lattice-free polyhedra which are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra $\\mathbb{Z}^d$-maximal.\n  It is known that, for fixed $d$, the family $\\mathbb{Z}^d$-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05200","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"}