Diagrams for Positive Bases
Diagrams for Positive Bases
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正碱基图
DOI：
10.1112/jlms/s2-4.1.165
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发表时间：
1971
影响因子：
1.2
通讯作者：
G. C. Shephard
中科院分区：
文献类型：
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作者：
G. C. Shephard
正基理论是研究凸性的一个有用工具。在本文中，我们用Gale图描述了有限维线性空间的正基的表示，它类似于多面体的表示（参见[4，§5.4和§6.3;7，第3章]）。基本思想隐含在Chandler Davis[2]的工作中，但是，由于我们现在在多面体研究中使用图技术的经验优势，我们能够以更强大的方式使用该方法。许多关于正基的基本结果，当用图表表示时，就变成了明显的几何命题。例如，如果X是rn的正基，则牌X < 2n（见[2；定理6.7]）在X的图中等价于每个a >多面体至少有k + l个顶点（见定理4）。再一次，我们将展示图表如何使我们以一种非常简单的方式，从X中挑选出它的所有子集，这些子集是张成最小子空间的。（关于“最小子空间”的定义见§2。）除了减少证明许多标准结果的劳动外，图表的使用使人们对正基的结构有了更深入的了解。在未来的研究中，这项技术可能会被证明是一个强有力的工具。我要感谢P. McMullen博士对本文早期版本的评论，特别是他让我注意到在证明Reay的定理（定理10）时使用了图表。
The theory of positive bases is a useful tool in the study of convexity. In this paper we describe a representation of a positive basis of a finite-dimensional linear space, which is analogous to that of a poly tope by a Gale diagram (see [4, §5.4 and §6.3; 7, Chapter 3]). The basic idea is implicit in the work of Chandler Davis [2] but, as we now have the advantage of experience in the use of diagram techniques in the study of polytopes, we are able to use the method in a much more powerful manner. Many of the basic results about positive bases, when expressed in terms of diagrams, become obvious geometric statements. For example, the fact that if X is a positive basis of R n , then card X < 2n (see [2; Theorem 6.7]) is equivalent, in the diagram of X, to the statement that every A>polytope has at least k + l vertices (see Theorem 4). Again, we shall show how a diagram enables us to pick out from X, in a very simple manner, all its subsets which span minimal subspaces. (For the definition of " minimal subspace " see §2.) Besides reducing the labour of proving many standard results, the use of a diagram gives much greater insight into the structure of a positive basis. It seems possible that this technique will prove a powerful tool in future investigations. I am indebted to Dr. P. McMullen for his comments on an early version of this paper, and, in particular, for drawing my attention to the use of diagrams in proving one of Reay's theorems (Theorem 10).