Random packings and coverings of the unit n-sphere
Random packings and coverings of the unit n-sphere
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单元 n 球体的随机堆积和覆盖物
DOI：
10.1002/j.1538-7305.1967.tb04246.x
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发表时间：
1967
影响因子：
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通讯作者：
A. Wyner
中科院分区：
文献类型：
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作者：
A. Wyner
众所周知，在n维欧氏空间中，单位球面上所能放置的最大不重叠的半角θ球帽数Mv（n，θ）不小于exp [-nlogsin 2θ + o（n）]（θ c（n，θ）），即覆盖n维欧氏球面所需的最小半角θ球帽数。证明了Mc（n，θ）= exp [-nlogsin θ + o（n）].证明的中心部分也是一个随机编码的论点，它断言如果随机选择一个近似exp（-n log sin θ）的帽集，平均来说，只有非常小的一部分n球表面将保持未覆盖（当n很大时）。
It is well known that the quantity M v (n, θ), the maximum number of nonoverlapping spherical caps of half angle θ (a “packing”) which can be placed on the surface of a unit sphere in Euclidean n-space is not less than exp [– n log sin 2θ + o(n)] (θ c (n, θ), the minimum number of caps of half angle θ required to cover the unit Euclidean n-sphere. We show that M c (n, θ) = exp [–n log sin θ + o(n)]. The central part of the proof is also a random coding argument which asserts that if a set roughly exp (–n log sin θ) caps is chosen at random, that on the average only a very small fraction of the surface of the n-sphere will remain uncovered (when n is large).