基于图论的球填充密度下界
A Lower Bound on the Density of Sphere Packings via Graph Theory
摘要：
利用图论方法，我们给出了一个新的证明，即对于所有足够大的$n$，存在密度至少为$cn2^{-n}$的$\R^n$中的球填充，超过经典Minkowski界的一个因子线性的$n$ .这符合一个常数最有名的下界的密度球包装由于罗杰斯，达文波特罗杰斯，和球。建议的方法使得有可能描述这样的包装与复杂性$\exp(n\log n)$，这是显着低于其他方法的点。
英文摘要：
Using graph-theoretic methods we give a new proof that for all sufficiently large $n$, there exist sphere packings in $\R^n$ of density at least $cn2^{-n}$, exceeding the classical Minkowski bound by a factor linear in $n$. This matches up to a constant the best known lower bounds on the density of sphere packings due to Rogers, Davenport-Rogers, and Ball. The suggested method makes it possible to describe the points of such a packing with complexity $\exp(n\log n)$, which is significantly lower than in the other approaches.
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