离散凸性与单模性。I
Discrete convexity and unimodularity. I
摘要：
本文建立了自由阿尔贝群M（整数点格）的凸性理论，我们称之为离散凸性理论。我们刻画了群M的那些子集X可以称为“凸”的。有一个性质似乎是无可争议的：X应该与它的凸包co（X）（在环境向量空间V中）的所有整数点的集合重合。然而，这是对适当的离散凸性的第一近似，因为这样的非相交集合不需要由超平面分开。这个问题与两个正多面体的交线是正多面体的问题密切相关。我们证明了幺模系统（或更一般地说，纯系统）与离散凸性类一一对应。例如，著名的g-多拟阵类对应于与Z^n中的幺模系统A_n：={ \pm e_i，e_i-ej}相关联的离散凸性类。
英文摘要：
In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property seems indisputable: X should coincide with the set of all integer points of its convex hull co(X) (in the ambient vector space V). However, this is a first approximation to a proper discrete convexity, because such non-intersecting sets need not be separated by a hyperplane. This issue is closely related to the question when the intersection of two integer polyhedra is an integer polyhedron. We show that unimodular systems (or more generally, pure systems) are in one-to-one correspondence with the classes of discrete convexity. For example, the well-known class of g-polymatroids corresponds to the class of discrete convexity associated to the unimodular system A_n:={\pm e_i, e_i-ej} in Z^n.
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