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# Convex-normal (Pairs of) Polytopes

Published:2016-11-15
Printed: Sep 2017
• Christian Haase,
Mathematik, FU Berlin , 14195 Berlin, Germany
• Jan Hofmann,
Mathematik, FU Berlin , 14195 Berlin, Germany
 Format: LaTeX MathJax PDF

## 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 $k\geq 3$) and improve the bound to $2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two 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 face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap \mathbb{Z}^d = (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.
 Keywords: integer decomposition property, integrally closed, projectively normal, lattice polytopes
 MSC Classifications: 52B20 - Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] 14M25 - Toric varieties, Newton polyhedra [See also 52B20] 90C10 - Integer programming

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