论文标题
通过离散减少的双线性编程界限用于球体包装
Dual Linear Programming Bounds for Sphere Packing via Discrete Reductions
论文作者
论文摘要
用于解决8和24维情况的球体包装的Cohn-Elkies线性程序猜想在任何其他维度上都不明确$ d> 2 $。通过将该无限维线性程序的可行点映射到通过离散减少的有限维问题中，我们提供了一种通用方法，以获取Cohn-Elkies线性程序的双重界限。这减少了变量的数量是有限的，从而可以应用计算机优化技术。使用此方法，我们证明Cohn-Elkies Bound不能接近尺寸已知的最佳包装密度$ 3 \ leq d \ leq 13 $，除了已解决的案例$ d = 8 $。特别是，我们的双重界限表明Cohn-Elkies结合无法解决3、4和5维球的包装问题。
The Cohn-Elkies linear program for sphere packing, which was used to solve the 8 and 24 dimensional cases, is conjectured to not be sharp in any other dimension $d>2$. By mapping feasible points of this infinite-dimensional linear program into a finite-dimensional problem via discrete reduction, we provide a general method to obtain dual bounds on the Cohn-Elkies linear program. This reduces the number of variables to be finite, enabling computer optimization techniques to be applied. Using this method, we prove that the Cohn-Elkies bound cannot come close to the best packing densities known in dimensions $3 \leq d \leq 13$ except for the solved case $d=8$. In particular, our dual bounds show the Cohn-Elkies bound is unable to solve the 3, 4, and 5 dimensional sphere packing problems.
