Representations of polytopes and polyhedral sets

In this paper we describe a variant of the diagram techniques, such as Gale diagrams for polytopes and positive diagrams for positive bases, which is appropriate for polyhedral sets. We obtain our new technique as a translation-invariant representation of polytopes or polyhedral sets. This approach leads naturally to simpler proofs of the familiar combinatorial diagram relationships. However, this method is more versatile than those previously employed, in that it can be used to investigate linear systems of polyhedral sets, and metrical properties such as volume. In particular, we give an easy proof of a result of Meyer on decomposability of polytopes, and a more perspicuous way of looking at the well-known theorem of Minkowski on the realizability of polytopes whose facets have given outer normal vectors and areas.

在本文中，我们描述了图表技术的一种变体，例如用于多面体的大风图和正面图的正图，这适用于多面体集合。我们将新技术作为多面体或多面体集的翻译不变表示。这种方法自然地导致了熟悉的组合图关系的更简单证明。但是，该方法比以前使用的方法更具用途，因为它可用于研究多面体集合的线性系统和诸如体积之类的度量属性。特别是，我们简单地证明了迈耶（Meyer）对多面体的可分解性的结果，以及一种更有明显的方式，即介绍了Minkowski的著名定理，这些定理对各个方面给出了外部正常向量和区域的多型的可靠性。