On Conjectures of Minkowski and Woods for n=9

TL;DR

This paper proves a conjecture in the geometry of numbers for nine-dimensional space, showing that certain lattice and sphere configurations satisfy longstanding conjectures by Minkowski and Woods.

Contribution

The authors establish the validity of Woods' conjecture and, consequently, Minkowski's conjecture in nine dimensions, extending known results from lower dimensions.

Findings

Woods' conjecture holds for n=9.

Minkowski's conjecture is confirmed for n=9.

The results build on previous work by McMullen (2005).

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 $n=9$.