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Classification of empty lattice 4-simplices of width larger than two

Abstract: A lattice $ d$-simplex is the convex hull of $ d+1$ affinely independent integer points in $ {\mathbb{R}}^d$. It is called empty if it contains no lattice point apart from its $ d+1$ vertices. The classification of empty $ 3$-simplices has been known since 1964 (White), based on the fact that they all have width one. But for dimension $ 4$ no complete classification is known. Haase and Ziegler (2000) enumerated all empty $ 4$-simplices up to determinant 1000 and based on their results conjectured that after determinant $ 179$ all empty $ 4$-simplices have width one or two. We prove this conjecture as follows: - We show that no empty $ 4$-simplex of width three or more can have a determinant greater than 5058, by combining the recent classification of hollow 3-polytopes (Averkov, Krümpelmann and Weltge, 2017) with general methods from the geometry of numbers. - We continue the computations of Haase and Ziegler up to determinant 7600, and find that no new $ 4$-simplices of width larger than two arise. In particular, we give the whole list of empty $ 4$-simplices of width larger than two, which is as computed by Haase and Ziegler: There is a single empty $ 4$-simplex of width four (of determinant 101), and 178 empty $ 4$-simplices of width three, with determinants ranging from 41 to 179.

 Fuente: Trans. Amer. Math. Soc. 371 (2019), 6605-6625

Editorial: American Mathematical Society

 Año de publicación: 2019

Nº de páginas: 21

Tipo de publicación: Artículo de Revista

 DOI: 10.1090/tran/7531

ISSN: 0002-9947,1088-6850

 Proyecto español: MTM2014-54207-P ; BES-2015-073128

Url de la publicación: https://doi.org/10.1090/tran/7531