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JCT
2011

Bounds for solid angles of lattices of rank three

13 years 6 months ago
Bounds for solid angles of lattices of rank three
We find sharp absolute constants C1 and C2 with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C1, C2]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in RN with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R3. Such spherical configurations come up in connection with the kissing number problem.
Lenny Fukshansky, Sinai Robins
Added 14 May 2011
Updated 14 May 2011
Type Journal
Year 2011
Where JCT
Authors Lenny Fukshansky, Sinai Robins
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