A note on the Besicovitch dimension of the closest packing of spheres in Rn
A note on the Besicovitch dimension of the closest packing of spheres in Rn
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关于 Rn 中球体最密堆积贝西科维奇维数的注解
DOI：
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发表时间：
1966
影响因子：
0.8
通讯作者：
D. Larman
中科院分区：
文献类型：
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作者：
D. Larman
导论.设sn表示真实的数集合X的下确界，其中x属于X当且仅当x是Rn的单位n-立方体In中的n-球面填充的剩余集的Besicovitch维数。在最近的工作（（1））中，我证明了s2大于1，很自然地，有人问我，这个证明是否可以推广到证明Rn中的类似结果sn大于n-1。虽然在理论上这是可以做到的，但实际上细节可能会变得相当复杂。然而，这样的推广是不必要的，因为结果sn > n − 1是将结果s2 > 1与以下定理结合的平凡结果。
Introduction. Let sn denote the infimum of the set of real numbers X, where x belongs to X if, and only if, x is the Besicovitch dimension of the residual set of a packing of n-spheres into the unit n-cube In, of Rn. In recent work ((1)) I have shown that s2 is greater than one, and, quite naturally, I have since been asked whether or not the proof can be generalized to prove the analogous result, sn greater than n − 1, in Rn. Whilst it is clear, in theory, that this could be done, in practice the details might become rather complicated. However, such a generalization is unnecessary, for the result sn > n − 1 is a trivial consequence of combining the result s2 > 1 with the following theorem.