Bounds for Solid Angles of Lattices of Rank Three

Authors

Lenny Fukshansky, Claremont McKenna CollegeFollow
Sinai Robins, Nanyang Technological University, Singapore

Document Type

Article

Department

Mathematics (CMC)

Publication Date

2-2011

Abstract

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.

Comments

More information on this article can be found at https://www.sciencedirect.com/science/article/pii/S0097316510000993?via%3Dihub

Rights Information

© 2011 Elsevier Inc.

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Terms of Use for work posted in Scholarship@Claremont.

DOI

http://dx.doi.org/10.1016/j.jcta.2010.06.001

Recommended Citation

Fukshansky, Lenny, and Sinai Robins. "Bounds for Solid Angles of Lattices of Rank Three." Journal of Combinatorial Theory, Series A 118.2 (2011): 690-701. Web.