New Conjectural Lower Bounds on the Optimal Density of Sphere Packings
New Conjectural Lower Bounds on the Optimal Density of Sphere Packings
复制标题
球填料最佳密度的新猜想下界
DOI：
--
复制
发表时间：
2005
影响因子：
0.5
通讯作者：
F. Stillinger
中科院分区：
文献类型：
--
作者：
S. Torquato;F. Stillinger
在高维度的球形填充感兴趣的数学家和物理学家，并在通信理论中有直接的应用。值得注意的是，由于闵可夫斯基在d维欧氏空间中的最大堆积密度的百年下界，没有人能够提供指数改进。这个界限的渐近行为是由2-d在高维控制。使用的优化过程，我们介绍了较早[Torquato和Stillinger 02]和一个猜想，关于存在的无序球填充在dropld，我们得到了一个几何下界的密度，其渐近行为控制2-0.77865. d，从而提供Minkowski界的假定指数改进。该猜想指出，一个硬核非负回火分布是一个对相关函数的几何不变的无序球包装在droxd渐近大的d当且仅当傅立叶变换的自协方差函数是非负的。该猜想是支持两个明确的分析特点无序包装，数字包装结构在低维，已知的必要条件，只有在非常低的维度，以及事实上，我们可以恢复的形式已知的严格的下限。我们的方法的一个副产品是平均接吻数的一个渐近的非线性下限，其行为由20.22134控制。d，这是比较最有名的渐近下限的个人接吻数20.2075. D .有趣的是，我们的优化过程正是Cohn和Elkies设计的原始线性规划的对偶[Cohn和Elkies 03]，以获得密度的上界，因此对线性规划界有影响。这种联系证明了我们的密度估计永远不会超过Cohn-Elkies上界，无论我们的猜想是否有效。
Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a hundred-year-old lower bound on the maximal packing density due to Minkowski in d-dimensional Euclidean space ℝ d . The asymptotic behavior of this bound is controlled by 2-d in high dimensions. Using an optimization procedure that we introduced earlier [Torquato and Stillinger 02] and a conjecture concerning the existence of disordered sphere packings in ℝd, we obtain a conjectural lower bound on the density whose asymptotic behavior is controlled by 2-0.77865... d , thus providing the putative exponential improvement of Minkowski's bound. The conjecture states that a hard-core nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in ℝ d for asymptotically large d if and only if the Fourier transform of the autocovariance function is nonnegative. The conjecture is supported by two explicit analytically characterized disordered packings, numerical packing constructions in low dimensions, known necessary conditions that have relevance only in very low dimensions, and the fact that we can recover the forms of known rigorous lower bounds. A byproduct of our approach is an asymptotic conjectural lower bound on the average kissing number whose behavior is controlled by 20.22134... d , which is to be compared to the best known asymptotic lower bound on the individual kissing number of 20.2075... d . Interestingly, our optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies [Cohn and Elkies 03] to obtain upper bounds on the density, and hence has implications for linear programming bounds. This connection proves that our density estimate can never exceed the Cohn– Elkies upper bound, regardless of the validity of our conjecture.