Geometry of the Minimal Solutions of a Linear Diophantine Equation

Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, let ${\mathcal S}$ denote the set of all nonnegative integer solutions equation $x_1a_1+\cdots +x_na_n=y_1b_1+\cdots...