Consider a full rank integer lattice in $\mathbb{R}^n$. Let $v_1$ be the shortest non-zero vector in the lattice, $v_2$ be the shortest one among those not parallel to $v_1$, $v_3$ be the shortest one not contained in the linear span of $v_1,\ v_2$, and so on until $v_n$. What is the infimum of the ratio of the volume of the parallelotope spanned by $v_1,...,v_n$ divided by $\prod_{i=1}^n|v_i|$, where the infimum is taken over all full rank lattices? Can the infimum be achieved, and if so, what are the extremal lattices?

When $n=2$, the infimal ratio is $\frac{\sqrt{3}}{2}$, uniquely achieved by the hexagonal lattice with $|v_1|=|v_2|$. The volume can be interpreted as the volume of a flat torus, and the $v_i$'s can be seen as closed geodesics spanning the fundamental group.