The sphere packing problem in dimension 8

Abstract

In this paper we prove that no packing of unit balls in Euclidean space $\mathbb{R}^8$ has density greater than that of the $E_8$-lattice packing.

Figures (2)

• Figure 1: Plot of the functions $A(t)$, $A^{(2)}_0(t)=-\frac{368640}{\pi^2}\,t^2\,e^{-\pi /t}$, and $A^{(1)}_\infty(t)=-\frac{72}{\pi^2}\,e^{2\pi t}+\frac{8640}{\pi}t-\frac{23328}{\pi^2}$.
• Figure 2: Plot of the functions $B(t)$, $B^{(2)}_0(t)=\frac{368640}{\pi^2}\,t^2\,e^{-\pi /t}$, and $B^{(1)}_\infty(t)=\frac{8640}{\pi}t-\frac{23328}{\pi^2}$.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4