Facing Up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes
Facing Up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes
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DOI：
10.1090/memo/0154
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发表时间：
1975-06
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影响因子：
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通讯作者：
T. Zaslavsky
中科院分区：
文献类型：
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作者：
T. Zaslavsky
欧氏或射影d-空间的超平面的排列是超平面的有限集合，连同空间的诱导划分。给定排列的超平面，如何计算诱导划分的面？在这里，这个问题已经回答了平面，欧几里德3-空间，超平面在一般位置，和d-面的超平面通过原点在欧几里德空间。在每一种情况下，k-面的数量仅取决于超平面的交叉点之间的入射，即使具有相同交叉点入射模式的排列通常不是组合同构的。我们推广这一事实，证明公式的k-面的所有欧几里德和射影安排，以及有界的k-面的前，作为功能的（半）格的交叉点的超平面，不依赖于安排的组合类型。
An arrangement of hyperplanes of Euclidean or projective d-space is a finite set of hyperplanes, together with the induced partition of the space. Given the hyperplanes of an arrangement, how can the faces of the induced partition be counted? Heretofore this question has been answered for the plane, Euclidean 3-space, hyperplanes in general position, and the d-faces of the hyperplanes through the origin in Euclidean space. In each case the numbers of k-faces depend only on the incidences between intersections of the hyperplane, even though arrangements with the same intersection incidence pattern are not in general combinatorially isomorphic. We generalize this fact by demonstrating formulas for the numbers of k-faces of all Euclidean and projective arrangements, and the numbers of bounded k-faces of the former, as functions of the (semi) lattice of intersections of the hyperplanes, not dependent on the arrangement's combinatorial type.