Distance Sums on a Sphere and Angle Sums in a Simplex
Distance Sums on a Sphere and Angle Sums in a Simplex
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球体上的距离和以及单纯形中的角度和
DOI：
10.1080/00029890.1956.11988764
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发表时间：
1956
期刊：
影响因子：
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通讯作者：
J. W. Gaddum
中科院分区：
文献类型：
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作者：
J. W. Gaddum
本文讨论关系(B)和(C)的推广。文献[1]的方法似乎不允许完全推广(B)，尽管它们给出了Sand的最佳上界和一个非平凡的下界。本文给出了S的最佳上下界。此外，(C)不是推广到一个方程，而是推广到一个不等式。我在[1]中说过，这些结果是可知的。似乎应该在这里说明它们是。(见[2])还应该指出的是，TS Motzkin在[3]中使用不同的方法对与本文相关的结果进行了调查。2.术语。在En中，两个(n-1)维超平面之间的二面角是指它们的有向法线之间的夹角的补充。给定En中的n个有向平面，其交点为点p，通过多面角，他们确定我们指的是他们在中心为p的单位球面S_n-1上切出的表面含量。
This paper is concerned with generalizing relations (B) and (C). The methods of [1] do not appear to permit a complete generalization of (B) although they give the best upper bound for Sand a non-trivial lower bound. This paper gives best upper and lower bounds for S. Furthermore,(C) does not generalize to an equation, but to an inequality.In [1] I remarked that one would expect those results to be known. It seems appropriate to state here that they were.(See [2].) It should also be mentioned that results related to those of this paper have been investigated by TS Motzkin in [3], using different methods. 2. Terminology. By the dihedral angle between two (n-1)-dimensional hyperplanes in En is meant the supplement of the angle between their directed normals. Given n directed planes in En whose intersection is a point p, by the multihedral angle they determine we mean the surface content they cut out on the unit sphere Sn-1 whose center is p. We suppose throughout that n> 2.