Narrow Results

We describe continuous increasing functions C_n(x) such that γ_n ≥ C_n(γ_n-1) where γ_m is Hermite's constant in dimension m. This inequality yields a new proof of the Minkowski–Hlawka bound Δ_n ≥ ζ(n)2^1-n for the maximal density Δ_n of n-dimensional lattice packings.

In the last half of 20th century, various generalizations of Hermite's constant and Voronoï's theorem were studied by many authors. In this paper, we give an account of a recent development concerning Voronoï's theorem.