On the optimal density of sphere packings in ${\mathbb{E}}^8$ and the uniqueness theorem on finite packings
with optimal density

Wu-Yi Hsiang

Abstract:Let ${\mathcal P}$ be a finite cluster of unit spheres and ${\mathcal P}^*$ be an extension of ${\mathcal P}$ which makes all the local cells (i.e. Voronoi cells) of $S_j\in {\mathcal P}$, $\{C(S_j,{\mathcal P}^*)\}$, bounded. Then, it is natural to define the relative density $\rho({\mathcal P},{\mathcal P}^*)$ by setting

$$\rho({\mathcal P},{\mathcal P}^*)={\textstyle \sum_{S_j\in {... ..._{S_j\in {\mathcal P}} }\, \hbox{vol}\, C(S_j,{\mathcal P}^*)$$ (1)

and to define the intrinsic density of ${\mathcal P}$ to be the least upper bound of $\{\rho({\mathcal P}, {\mathcal P}^*)\}$ for all possible extensions of ${\mathcal P}$, namely

$$\rho({\mathcal P})=l.u.b.\, \{\rho({\mathcal P}, {\mathcal P}^*), {\mathcal P}\subset {\mathcal P}^*\}$$ (2)

In the very special case that ${\mathcal P}=\{S\}$ consists of a single sphere, the above intrinsic density $\rho(\{S\})$ is exactly the optimal local density of sphere packings in ${\mathbb{E}}^8$. Anyhow, the following are two fundamental problems in the study of sphere packings in ${\mathbb{E}}^8$, namely

Problem 1:What is the optimal local density in ${\mathbb{E}}^8$? and what are the geometric structures of those tightest local packings (i.e. the ones with optimal local density) ?

Problem 2:What is the optimal intrinsic density for cluster of unit spheres in ${\mathbb{E}}^8$ of a given cardinality, namely

$$\rho_N=l.u.b.\, \{\rho({\mathcal P}),\char93 ({\mathcal P})=N\}=\,?$$ (3)

Moreover, what are the geometric structures of those $N$-clusters with $\rho({\mathcal P})=\rho_N$ ?

It is a remarkable, pleasant surprise that both of the above two problems have clean-cut solutions and the strongest possible uniqueness theorems, namely

Theorem I:The optimal local density of sphere packings in ${\mathbb{E}}^8$ is equal to $\pi^4/384$. The local density of a local packing ${\mathcal L}(S_0)$ is equal to $\pi^4/384$ when and only when ${\mathcal L}(S_0)$ is isometric to the local packing type of the lattice packing associated to the root lattice of $E_8$ (i.e. the exceptional Lie group of rank 8).

Theorem II:Let ${\mathcal P}$ be a finite cluster of identical spheres in ${\mathbb{E}}^8$. Then

$$\rho({\mathcal P})\leq \frac{\pi^4}{384}$$ (4)

and equality holds when and only when ${\mathcal P}$ is isometric to a finite cluster in the $E_8$-lattice packing (up to a scaling). [A finite packing is called a cluster if any pair of them can be linked by a chain with consecutive center-distances less than $2\sqrt{2}$-time of the radius.]

The purpose of this talk is to present the proofs of the above two theorems.
[In fact, Theorem II follows readily from Theorem I.]