A Lower Bound on the Number of Cells in Arrangements of Hyperplanes
A Lower Bound on the Number of Cells in Arrangements of Hyperplanes
复制标题
超平面排列中单元数量的下界
DOI：
10.1016/0097-3165(76)90027-3
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发表时间：
1976
期刊：
影响因子：
--
通讯作者：
R. W. Shannon
中科院分区：
文献类型：
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作者：
R. W. Shannon
本文的主要结果是定理4，它给出了Pd中y ~ 1超平面排列中k-胞元个数的一个下界。对于k= 0，这已经由Motzkin [6]，Hanani [4]，Basterfield和Kelly [2]建立。这个结果在d= 2时是众所周知的;对于k= d，McMullen在1971年证明了这个结果（未发表），而Canham [3]证明了k= d-1和k= d的情况。一般猜想首先由格伦鲍姆提出[7，p.3931]。为了得到定理4，我们利用高维排列中普通顶点的存在。这种顶点的存在性（或者说对偶结果）由Motzkin [61]证明，并由他建立了d= 3。Balomenos，Bonnice和Silverman [1]证明了Motzkin猜想对于d< 5，汉森[5]建立了它对于所有d。我们将给出汉森定理的一个稍微不同的公式和简化的证明。在建立了n个超平面的排列中的k-单元数目的下界之后，我们继续描述获得下界的排列。这已经由Basterfield和Kelly [2]在k= 0的情况下完成了（在对偶设置中）。
The main result in this paper is Theorem 4, which gives a lower bound on the number of k-cells in an arrangement of y1 hyperplanes in Pd. For k= 0 this has already been established by Motzkin [6], Hanani [4], and Basterfield and Kelly [2]. The result is well-known for d= 2; for k= d it was proved by McMullen in 1971 (unpublished), while Canham [3] proved the cases k= d-1 and k= d. The general conjecture was first proposed by Grunbaum [7, p. 3931. To obtain Theorem 4 we utilize the presence of ordinary vertices in higher-dimensional arrangements. The existence of such vertices (or rather the dual result) was conjectured by Motzkin [61 and established by him for d= 3. Balomenos, Bonnice, and Silverman [l] proved Motzkin’s conjecture for d< 5 and Hansen [5] established it for all d. We shall give a somewhat different formulation and simplified proof of Hansen’s theorem. After establishing the lower bounds for the numbers of k-cells in arrangements of n hyperplanes, we proceed to characterize the arrangements in which the lower bounds are obtained. This has been done (in the dual setting) for the case k= 0 by Basterfield and Kelly [2].