2012/06/20 by Danny Nguyen, Nguyen, Danny
Computer Science · Mathematics · #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1206.4390
openalex publication_date 2012/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a lattice Λ⊂ ℝn, we consider its Minkowski reduced basis and the solid angle Ω 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 Ω. Sharp upper and lower bounds are derived for all rank 3 and rank 4 lattices so that Ω always measures in between. Extreme cases happen when Λ is similar to the rectangular (R) or alternating (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 Ω as a product of det(Λ) and a quadratic integral on the unit sphere \mathbbSn-1. Finally, we show that for rank 5, the alternating lattice A5 no longer possesses the smallest measure for Ω.