CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

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 integer decomposition property, integrally closed, projectively normal, lattice polytopes
MSC Classifications: 52B20, 14M25, 90C10 show english descriptions Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]
Toric varieties, Newton polyhedra [See also 52B20]
Integer programming
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
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/