On conjectures of Minkowski and Woods for $n=10$

Abstract

Let $\mathbb{L}$ be a lattice in $n$-dimensional Euclidean space $\mathbb{R}^n$ reduced in the sense of Korkine and Zolotareff and having a basis of the form $~(A_1,0,0,\cdots$ $,0),$ ~$(a_{2,1},A_2,0,\cdots,0),\cdots,$ $(a_{n,1},a_{n,2},\cdots,a_{n,n-1},A_n)$. A famous conjecture of Woods in Geometry of Numbers asserts that if $A_1A_2\cdots A_n = 1$ and $A_i\leq A_1$ for each $i$ then any closed sphere in $\mathbb{R}^n$ of radius $\sqrt{n/4}$ contains a point of $\mathbb{L}.$ Together with a result of C. T. McMullen (2005), the truth of Woods' Conjecture for a fixed $n$, implies the long standing classical conjecture of Minkowski on product of $n$ non-homogeneous linear forms for that value of $n$. In an earlier paper `Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, 2016, 501-548' we proved Woods' Conjecture for $n=9$. In this paper, we prove Woods' Conjecture and hence Minkowski's Conjecture for $n=10$.