Solid angles associated to Minkowski reduced bases

Abstract

Given a lattice $\Lambda \subset \mathbb{R}^n$, we consider its Minkowski reduced basis and the solid angle $\Omega$ spanned by the basis vectors. Such a basis satisfies strong near-orthogonality conditions, which allow us to bound from above and below the measure of $\Omega$. Sharp upper and lower bounds are derived for all rank $3$ and rank $4$ lattices so that $\Omega$ always measures in between. Extreme cases happen when $\Lambda$ is similar to the rectangular ($\mathcal{R}$) or alternating ($\mathcal{A}$) lattice. This result settles a question raised earlier by Fukshansky and Robins in connection to sphere packings and kissing numbers. The proof relies on a formula by Hajja and Walker that expresses $\Omega$ as a product of $\det(\Lambda)$ and a quadratic integral on the unit sphere $\mathbb{S}^{n-1}$. Finally, we show that for rank 5, the alternating lattice $\mathcal{A}_{5}$ no longer possesses the smallest measure for $\Omega$.