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arXiv · 1410.6430

Convex-normal (pairs of) polytopes

Christian Haase·Jan Hofmann
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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) \cap \Z^d = (P\cap \Z^d) + (Q \cap \Z^d).

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Christian Haase, Jan Hofmann. 2014-10-23. Convex-normal (pairs of) polytopes. https://arxiv.org/abs/1410.6430

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