Convex Polytopes: Extremal Constructions and f -Vector Shapes
Convex Polytopes: Extremal Constructions and f -Vector Shapes
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凸多面体：极值结构和 f 向量形状
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发表时间：
2004
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通讯作者：
G. Ziegler
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作者：
G. Ziegler
在过去的四十年里，f向量的研究取得了巨大的成功.最基本的一个定理无疑是由McMullen在1971年提出并由Billera & Lee和Stanley在1980年证明的“g-定理”，它组合地刻画了单多面体和单多面体的f向量。另见第5.2节福尔曼的文章在这卷，其中h-向量讨论了与查尼-戴维斯猜想。然而，在一些基本问题令人尴尬的小进展，其中一个值得注意的问题涉及的形状f -向量的4-多面体。许多引人注目的和迷人的多面体结构已提出和分析多年。特别地，Billera-Lee构造产生单纯多面体的“所有可能的f -向量”。不太明显的进展是在简单或单纯多面体的范围外取得的-我们的进展的衡量标准是，新的多面体“有趣的f -向量”应该产生。因此，仍然“似乎总的来说，我们缺乏例子。提出数学中有用的例子（或通常认为的反例）的方法甚至比证明数学陈述的方法更不清楚”（吉尔·卡莱，2000）。这些课堂讲稿旨在展示这两个研究领域的富有成效的相互作用：对f -向量形状的讨论提出了“极值”多面体的概念，即具有“极值f -向量形状”的多面体。我们在这里讨论的构造的选择是由以下指导的：我们将研究产生有趣的f -向量形状的构造。
The study of f -vectors has had huge successes in the last forty years. The most fundamental one is undoubtedly the “g-theorem,” conjectured by McMullen in 1971 and proved by Billera & Lee and Stanley in 1980, which characterizes the f vectors of simplicial and of simple polytopes combinatorially. See also Section 5.2 of Forman’s article in this volume, where h-vectors are discu ssed in connection with the Charney–Davis conjecture. Nevertheless, on some fundamental problems embarassingly little progress was made; one notable such problem concerns the shapes of f -vectors of 4-polytopes. A number of striking and fascinating polytope constructions have been proposed and analyzed over the years. In particular, the Billera–Lee construction produces “all possible f -vectors” of simplicial polytopes. Less visible progress was made outside the range of simple or simplicial polytopes — where our measure of progress is that new polytopes “with interesting f -vectors” should be produced. Thus, still “it seems that overall, we are short of examples. The methods for coming up with useful examples in mathematics (or counterexamples to commonly believed conjectures) are even less clear than the methods for proving mathematical statements” (Gil Kalai, 2000). These lecture notes are meant to display a fruitful interplay of these two areas of study: The discussion of f -vector shapes suggests the notion of “extremal” polytopes, that is, of polytopes with “extremal f -vector shapes.” Our choice of constructions to be discussed here is guided by this: We will be looking at constructions that produce interesting f -vector shapes.