On conjectures of Minkowski and Woods for $\varvec{n=10}$

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 ($A_1,0,0,\ldots ,0$), $(a_{2,1},A_2,0,\ldots ,0),\ldots ,(a_{n,1},a_{n,2},\ldots ,a_{n,n-1},A_n)$. A famous conjecture of Woods in geometry of numbers asserts that if $A_1A_2\ldots A_n = 1$ and $A_i\le A_1$ 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 McMullen (J. Am. Math. Soc. 18  (2005) 711–734), 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.) 126  (2016) 501–548), the authors proved Woods’ conjecture for $n=9$. In this paper, we prove Woods’ conjecture and hence Minkowski’s conjecture for $n=10$.