Convex-normal (pairs of) polytopes

2014/10/23 by Haase, Christian, Hofmann, Jan · 1 citation
#52B20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1410.6430

Abstract

In 2012 Gubeladze (Adv. Math. 2012) introduced the notion of k-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 difference between k- and (k+1)-convex-normality (for k >= 3) and improve the bound to 2d(d+1). In the second part we extend the definition to pairs of polytopes and show that for rational polytopes P and Q, where the normal fan of P is a refinement of the normal fan of Q, if every edge e_P of P is at least d times as long as the corresponding edge e_Q of Q, then (P+Q) ∩ \Z^d = (P∩ \Z^d) + (Q ∩ \Z^d).

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• Is there a smooth lattice polytope which does not have the integer decomposition property?

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