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Optim."],"published-print":{"date-parts":[[2024,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Given a rational pointed [Formula: see text]-dimensional cone [Formula: see text], we study the integer Carath\u00e9odory rank [Formula: see text] and its asymptotic form [Formula: see text], where we consider \u201cmost\u201d integer vectors in the cone. The main result significantly improves the previously known upper bound for [Formula: see text]. We also study bounds on [Formula: see text] in terms of [Formula: see text], the maximal absolute [Formula: see text] minor of the matrix given in an integral polyhedral representation of [Formula: see text]. If [Formula: see text], we show that [Formula: see text], and prove upper bounds for simplicial cones, improving the best known upper bound on [Formula: see text] for [Formula: see text].<\/jats:p>","DOI":"10.1137\/23m1561312","type":"journal-article","created":{"date-parts":[[2024,1,9]],"date-time":"2024-01-09T04:22:22Z","timestamp":1704774142000},"page":"190-200","source":"Crossref","is-referenced-by-count":3,"title":["New Bounds for the Integer Carath\u00e9odory Rank"],"prefix":"10.1137","volume":"34","author":[{"given":"Iskander","family":"Aliev","sequence":"first","affiliation":[{"name":"School of Mathematics, Cardiff University, Cardiff, Wales, CF24 4AG, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Henk","sequence":"additional","affiliation":[{"name":"Institute for Mathematics, Technische Universit\u00e4t Berlin, Berlin, 10623, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Hogan","sequence":"additional","affiliation":[{"name":"School of Mathematics, Cardiff University, Cardiff, Wales, CF24 4AG, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan","family":"Kuhlmann","sequence":"additional","affiliation":[{"name":"Department of Mathematics, ETH Z\u00fcrich, Z\u00fcrich, 8092, Switzerland."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Timm","family":"Oertel","sequence":"additional","affiliation":[{"name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg, 91054 Erlangen, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,1,9]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(01)00337-7"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-021-01657-8"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/16M1083876"},{"key":"ref4","doi-asserted-by":"crossref","unstructured":"S. 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