All types implies torsion

Abstract:

We prove the following theorem. Given a positive integer $n$ and a subset $A$ of ${{\mathbf {Z}}^n}$ with the following properties: (1) for every $\left ( {{a_1}, \ldots ,{a_n}} \right )$ in $A$ the inequality $\Sigma _{i = 1}^n {\left | {{a_i}} \right |} \geq 2$ holds, and (2) for every $\left ( {{x_1}, \ldots ,{x_n}} \right )$ in ${{\mathbf {R}}^n}$ there exists an $\left ( {{a_1}, \ldots ,{a_n}} \right )$ in $A$ with ${a_i}{x_i} \geq 0$ for $i = 1, \ldots ,n$, there exists a subset ${A_0}$ of $A$ such that ${{\mathbf {Z}}^n}$ modulo the subgroup generated by ${A_0}$ contains a nontrivial torsion element.