Affine Subdivision, Steerable Semigroups, and Sphere Coverings
Affine Subdivision, Steerable Semigroups, and Sphere Coverings
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DOI：
10.4310/pamq.2007.v3.n4.a2
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发表时间：
2007
影响因子：
0.7
通讯作者：
R. Schwartz
中科院分区：
文献类型：
--
作者：
R. Schwartz
设n是一个欧氏n-单形，{n}表示分割n的单形的有限并。我们假设该划分在仿射对称群下是不变的。这种划分的一个经典例子是从重心细分获得的划分，但还有很多其他可能性。(See（见第4.1节，或关于重心细分的定义。）我们的分区产生了仿射细分规则的，这可能是迭代。为了细分每个j，我们选择一个仿射映射Aj，其中j = Aj（i），然后将j划分为集合{Aj（i）}。分割的仿射不变性转化为这样一个事实，即我们的分割是独立的（n+ 1）！不同的选择J。现在我们来谈谈。一个基本的问题，人们可以问的是，迭代的细分规则产生一个密集的一组形状的单纯形？通过单形的形状，我们指的是被认为模相似的单形。在[英国广播公司]提出了这个问题，并回答肯定的情况下，2维重心细分。在[S]中，我们在3维中得到了相同的结果。一般来说，对于刚才提到的那种密度结果的第一步如下：设Cn是通过迭代应用细分规则获得的所有n维单形的集合。设Ωn是形式为± L的矩阵的集合，|det（L）|1/n，
Let ∆ be a Euclidean n-simplex and let {∆j} denote a finite union of simplices which partition ∆. We assume that the partition is invariant under the affine symmetry group of ∆. A classical example of such a partition is the one obtained from barycentric subdivision, but there are plenty of other possibilities. (See §4.1, or else [Sp, p. 123], for a definition of barycentric subdivision.) Our partition gives rise to an affine subdivision rule for ∆, which may be iterated. To subdivide each ∆j , we choose an affine map Aj with ∆j = Aj(∆), and then partition ∆j into the collection {Aj(∆i)}. The affine invariance of the partition translates into the fact that our partition of ∆j is independent of the (n+ 1)! different choices for Aj . Now we iterate. A basic question one can ask is Does the iteration of the subdivision rule produce a dense set of shapes of simplices? By shape of a simplex, we mean a simplex considered mod similarities. In [BBC] this question was raised and answered affirmatively for the case of 2-dimensional barycentric subdivision. In [S] we got the same result in 3 dimensions. In general, a first step for the kind of density results just mentioned is as follows: Let Cn be the collection of all n-dimensional simplices obtained by iteratively applying the subdivision rule. Let Ωn be the collection of matrices of the form ± L | det(L)|1/n ,