Dual linear programming bounds for sphere packing via discrete reductions

IF 1.5 1区 数学 Q1 MATHEMATICS Advances in Mathematics Pub Date : 2024-11-27 DOI:10.1016/j.aim.2024.110043
Rupert Li
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Abstract

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 3d13 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.
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通过离散还原实现球体包装的双重线性规划边界
用于求解 8 维和 24 维情况的球体包装的 Cohn-Elkies 线性程序,据猜测在任何其他维度 d>2 下都不尖锐。通过离散还原法将这个无限维线性程序的可行点映射为有限维问题,我们提供了一种获得 Cohn-Elkies 线性程序对偶约束的通用方法。这就将变量的数量减少到有限,使计算机优化技术得以应用。利用这种方法,我们证明了除了 d=8 的求解情况外,Cohn-Elkies 边界无法接近维数 3≤d≤13 的已知最佳堆积密度。特别是,我们的对偶边界表明,Cohn-Elkies 边界无法解决 3 维、4 维和 5 维球体堆积问题。
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来源期刊
Advances in Mathematics
Advances in Mathematics 数学-数学
CiteScore
2.80
自引率
5.90%
发文量
497
审稿时长
7.5 months
期刊介绍: Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.
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