A new inequality for the Hermite constants

Abstract

We prove an inequality of the form ${\gamma }_n\ge C_n({\gamma }_{n-1})$ giving a lower bound for the Hermite constant ${\gamma }_n$ in dimension $n$ in terms of ${\gamma }_{n-1}$. Iterating this inequality yields a new proof of the Minkowski-Hlawka bound.

24 pages, to appear in Internation Journal of Number Theory