Discrete convexity and unimodularity—I
Discrete convexity and unimodularity—I
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离散凸性和单模性——I
DOI：
10.1016/j.aim.2003.11.010
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发表时间：
2003
影响因子：
1.7
通讯作者：
G. Koshevoy
中科院分区：
文献类型：
--
作者：
V. Danilov;G. Koshevoy
在本文中，我们发展了一个理论的凸性的整数点格Zn，我们称之为理论的离散凸性。也就是说，我们的特征类的子集的Zn，它具有分离属性，或等价地，类的整数多面体，使任何两个多面体的类的交集是一个整数多面体（不需要在类）。具体来说，我们表明，这些（最大）类是在一对一的对应与纯系统。幺模系统是纯系统的一个重要例子。给定一个幺模系统，我们构造了一对（对偶）离散凸类，其中一个在求和下稳定，另一个在交下稳定。
In this paper we develop a theory of convexity for the lattice of integer points Zn, which we call theory of discrete convexity. Namely, we characterize classes of subsets of Zn, which possess the separation property, or, equivalently, classes of integer polyhedra such that intersection of any two polyhedra of a class is an integer polyhedron (need not be in the class). Specifically, we show that these (maximal) classes are in one-to-one correspondence with pure systems. Unimodular systems constitute an important instance of pure systems. Given a unimodular system, we construct a pair of (dual) discretely convex classes, one of which is stable under summation and the other is stable under intersection.