# Minimal volume of fundamental domains of lattices

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.

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