# Lattices achieving best density

Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ is the packing radius of $\Lambda$ (i.e., $\rho = (1/2) \displaystyle \min_{0 \neq x \in \Lambda} \left\| x \right\|$). Let $\delta_n$ be the supremum of $\delta(\Lambda)$ over all lattices in $\mathbb{R}^n$. Mahler's Selection Theorem implies that there is a lattice whose density achieves $\delta_n$ (see e.g. [Lekkerkeker and Gruber, Geometry of Numbers], pp. 260).

My question is: what is the most elementary way of proving the fact that there exists a lattice achieving the best packing density, without using any Selection Theorem (is there any)?

• Not exactly elementary, but the Voronoi theory of perfect quadratic forms is another usual route to define the densest lattice packing, and does not usually invoke Mahler's selection principle. See for example: books.google.com/… Dec 18 '13 at 2:15
• Also, Hermite defined the Hermite constant (equivalent up to some factors to $\delta_n$) way before Mahler came around. Maybe it's worth tracking the original reference? Dec 18 '13 at 2:20
• The oldest reference I could find on the internet (this was fun!) was Hermite original letter "Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objects de la théorie des nombre" [link](gdz.sub.uni-goettingen.de/dms/load/img/… ). As far as my french can go, I don't think he deals with these subtleties. Dec 18 '13 at 3:32