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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.

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I just notice that Minkowski's second theorem provides an upper and lower bound for the volume of the fundamental domain: $\frac{2^n}{n!}vol(\Lambda)\leq \lambda_1...\lambda_n\omega_n\leq 2^n vol(\Lambda)$,

where $\Lambda$ is the lattice, vol is its volume, $\omega_n$ is the volume of the n-dim unit ball, and $\lambda_i$'s are the successive minima of the lattice.

https://encyclopediaofmath.org/wiki/Minkowski_theorem

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