{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T00:49:51Z","timestamp":1787359791513,"version":"3.56.0"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Let $\\mathcal{T}^d$ be the set of all d-dimensional simplices T in $\\mathbb{R}^d$ with integer vertices and a single integer point in the interior of T. It follows from a result of Hensley that $\\mathcal{T}^d$ is finite up to affine transformations that preserve $\\mathbb{Z}^d$. It is known that when d grows, the maximum volume of the simplices $T \\in \\mathcal{T}^d$ becomes extremely large. We improve and refine bounds on the size of $T \\in \\mathcal{T}^d$ (where by the size we mean the volume or the number of lattice points). It is shown that each $T \\in \\mathcal{T}^d$ can be decomposed into an ascending chain of faces $G_1 \\subseteq \\cdots \\subseteq G_d=T$ such that for every $i \\in \\{1,\\ldots,d\\}$, $G_i$ is i-dimensional and the size of $G_i$ is bounded from above in terms of i and d. The bound on the size of $G_i$ is double exponential in i. The presented upper bounds are asymptotically tight on the log-log scale.<\/jats:p>","DOI":"10.1137\/110829052","type":"journal-article","created":{"date-parts":[[2012,4,26]],"date-time":"2012-04-26T18:20:00Z","timestamp":1335464400000},"page":"515-526","source":"Crossref","is-referenced-by-count":9,"title":["On the Size of Lattice Simplices with a Single Interior Lattice Point"],"prefix":"10.1137","volume":"26","author":[{"given":"Gennadiy","family":"Averkov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,4,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1110.0510"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"A. Barvinok,\n                      A Course in Convexity\n                      , Grad. Stud. Math. 54, American Mathematical Society, Providence, RI, 2002.","DOI":"10.1090\/gsm\/054"},{"key":"R3","first-page":"134","volume":"183","author":"Borisov A. A.","year":"1992","journal-title":"Mat. Sb.","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"key":"R4","unstructured":"A. Borisov,\n                      Convex lattice polytopes and cones with few lattice points inside, from a birational geometry viewpoint\n                      , arXiv:math\/0001109, 2000."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.2307\/2299023"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"A. Del Pia and R. Weismantel,\n                      On convergence in mixed integer programming\n                      , Math. Program. Ser. A, to appear, DOI: 10.1007\/s10107-011-0476-9.","DOI":"10.1007\/s10107-011-0476-9"},{"key":"R7","first-page":"483","volume":"240","author":"Ehrhart E.","year":"1955","journal-title":"C. R. Acad. Sci. 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Lawrence,\n                      \n                        Finite unions of closed subgroups of the\n                        n\n                        -dimensional torus\n                      \n                      , in Applied Geometry and Discrete Mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 4, American Mathematical Society, Providence, RI, 1991, pp. 433\u2013441.","DOI":"10.1090\/dimacs\/004\/34"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1991-058-4"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1110.0503"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1112\/S0025579300014339"},{"key":"R20","first-page":"83","volume":"28","author":"Rabinowitz S.","year":"1989","journal-title":"Ars Combin.","ISSN":"https:\/\/id.crossref.org\/issn\/0381-7032","issn-type":"print"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"R. Schneider,\n                      Convex Bodies: The Brunn-Minkowski Theory\n                      , Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, UK, 1993.","DOI":"10.1017\/CBO9780511526282"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700022826"},{"key":"R23","first-page":"44","volume":"37","author":"Zaks J.","year":"1982","journal-title":"Elem. 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