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 and 4 dimensional sphere packing problems.
翻译:用于解决8和24维案例的Cohn-Elkies球体包装线性程序被推断为在任何其他维度中不会是尖锐的 $d>2$。通过将这个无限维线性程序可行的点绘制成一个通过分解减少的有限维度问题,我们提供了一种一般方法,以便在Cohn-Elkies线性程序上获得双重界限。这减少了有限变量的数量,使计算机优化技术得以应用。使用这种方法,我们证明Cohn-Elkies捆绑无法接近在3\leq d\leq 13$这一维度中已知的最佳包装密度,但已解决的个案除外。特别是,我们的两维界限显示Cohn-Elkies线性程序无法解决3和4维区域包装问题。