On Conjectures of Minkowski and Woods for n=9

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 $\mathrm{\mid }X\mathrm{\mid }<R$ 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 $\sqrt{n\mathrm{/}4}$ contains a point of $\wedge$. This is known to be true for $n\le 8$. 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 $n=9$.