Abstract
AbstractIn 2012, Gubeladze (Adv. Math. 2012) introduced the notion ofk-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+ 1) have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence betweenk- and (k+ 1)-convex-normality (fork≥ 3) and improve the bound to 2d(d+ 1). In the second part we extend the definition to pairs of polytopes. Given two rational polytopesPandQ, where the normal fan ofPis a reûnement of the normal fan ofQ, if every edge ePofPis at least d times as long as the corresponding face (edge or vertex) eQofQ, then.
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