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The finiteness threshold width of lattice polytopes

Abstract: In each dimension d there is a constant woo(d) [épsilon] N such that for every n [épsilon] N all but finitely many lattice d-polytopes with n lattice points have llattice width at most woo(d). We call woo(d) the finiteness threshold width in dimension d and show that d - 2 woo(d) O _d4/3_. Blanco and Santos determined the value woo(3) = 1. Here, we establish woo(4) = 2. This implies, in particular, that there are only finitely many empty 4-simplices of width larger than two. (This last statement was claimed by Barile et al. in [Proc. Am. Math. Soc. 139 (2011), pp. 4247-4253], but we have found a gap in their proof.) Our main tool is the study of d-dimensional lifts of hollow (d-1)-polytopes.

 Fuente: Transactions of the American Mathematical Society, Series. B, 2021, 8, 399-419

 Editorial: American Mathematical Society

 Fecha de publicación: 01/04/2021

 Nº de páginas: 21

 Tipo de publicación: Artículo de Revista

 DOI: 10.1090/btran/62

 ISSN: 0002-9947,1088-6850

 Proyecto español: MTM2014-54207-P