On Conjectures of Minkowski and Woods for n=9

Abstract

Let $\mathbb{R}^n$ be the n-dimensional Euclidean space with $O$ as the origin. Let $\wedge$ be a lattice of determinant $1$ such that there is a sphere $|X|<R$ which contains no point of $\wedge$ other than $O$ and has $n$ linearly independent points of $\wedge$ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in $\mathbb{R}^n $ of radius $ \sqrt{n/4}$ contains a point of $\wedge$. This is known to be true for $n\leq 8$. Here we prove a more general conjecture of Woods for $n=9$ from which this conjecture follows in $\mathbb{R}^9$. Together with a result of C. T. McMullen (2005), the long standing conjecture of Minkowski follows for $n=9$.