{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T06:38:12Z","timestamp":1772951892978,"version":"3.50.1"},"reference-count":26,"publisher":"Institute for Operations Research and the Management Sciences (INFORMS)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics of OR"],"published-print":{"date-parts":[[2017,11]]},"abstract":"<jats:p> Lattice-free sets and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. The family of all integral lattice-free polyhedra that are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra \u2124<jats:sup>d<\/jats:sup>-maximal. <\/jats:p><jats:p> For fixed d, the family of \u2124<jats:sup>d<\/jats:sup>-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cutting-plane theory, one would like to have a classification of this family. This is a challenging task already for small dimensions. <\/jats:p><jats:p> In contrast, the subfamily of all integral lattice-free polyhedra that are not properly contained in any other lattice-free set, which we call \u211d<jats:sup>d<\/jats:sup>-maximal lattice-free polyhedra, allow a rather simple geometric characterization. Hence, the question was raised for which dimensions the notions of \u2124<jats:sup>d<\/jats:sup>-maximality and \u211d<jats:sup>d<\/jats:sup>-maximality are equivalent. This was known to be the case for dimensions one and two. On the other hand, for d \u2265 4 there exist integral lattice-free polyhedra that are \u2124<jats:sup>d<\/jats:sup>-maximal but not \u211d<jats:sup>d<\/jats:sup>-maximal. We consider the remaining case d = 3 and prove that for integral lattice-free polyhedra the notions of \u211d<jats:sup>3<\/jats:sup>-maximality and \u2124<jats:sup>3<\/jats:sup>-maximality are equivalent. This allows to complete the classification of all \u2124<jats:sup>3<\/jats:sup>-maximal integral lattice-free polyhedra. <\/jats:p>","DOI":"10.1287\/moor.2016.0836","type":"journal-article","created":{"date-parts":[[2017,4,18]],"date-time":"2017-04-18T18:29:16Z","timestamp":1492540156000},"page":"1035-1062","source":"Crossref","is-referenced-by-count":17,"title":["Notions of Maximality for Integral Lattice-Free Polyhedra: The Case of Dimension Three"],"prefix":"10.1287","volume":"42","author":[{"given":"Gennadiy","family":"Averkov","sequence":"first","affiliation":[{"name":"Otto-von-Guericke-Universit\u00e4t Magdeburg, 39106 Magdeburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jan","family":"Kr\u00fcmpelmann","sequence":"additional","affiliation":[{"name":"Otto-von-Guericke-Universit\u00e4t Magdeburg, 39106 Magdeburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Weltge","sequence":"additional","affiliation":[{"name":"ETH Z\u00fcrich, 8092 Z\u00fcrich, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"109","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1137\/080744360"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-72792-7_1"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1007\/s13366-012-0092-8"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1007\/s13366-011-0028-8"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1110.0510"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/054"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-009-0281-x"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1016\/j.sorms.2011.03.001"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01580858"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1110.0527"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-011-0476-9"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-010-0362-x"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-89597-4.50005-6"},{"key":"B14","series-title":"North-Holland Mathematical Library","volume-title":"Geometry of Numbers.","volume":"37","author":"Gruber PM","year":"1987","edition":"2"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(90)90010-A"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.2307\/1971436"},{"key":"B17","series-title":"Mathematics and Its Applications (Japanese Series)","first-page":"177","volume-title":"Mathematical Programming: Recent Developments and Applications.","volume":"6","author":"Lov\u00e1sz L","year":"1989"},{"key":"B18","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1110.0503"},{"key":"B19","first-page":"83","volume":"28","author":"Rabinowitz S","year":"1989","journal-title":"Ars Combin."},{"key":"B20","series-title":"Princeton Landmarks in Mathematics","volume-title":"Convex Analysis.","author":"Rockafellar RT","year":"1997"},{"key":"B21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01899997"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1287\/moor.10.3.403"},{"key":"B23","series-title":"Encyclopedia of Mathematics and its Applications (No. 151)","volume-title":"Convex Bodies: The Brunn-Minkowski Theory.","author":"Schneider R","year":"2014","edition":"2"},{"key":"B24","series-title":"Wiley-Interscience Series in Discrete Mathematics","volume-title":"Theory of Linear and Integer Programming.","author":"Schrijver A","year":"1986"},{"key":"B26","unstructured":"Treutlein J (2010) Birationale Eigenschaften generischer Hyperfl\u00e4chen in algebraischen Tori. 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