Minkowski's second theorem

Remarks

The lower bound is elementary: $K$ contains the cross-polytope spanned by $\pm v_i/\lambda_i$ for suitable independent lattice vectors $v_i$, and this cross-polytope has volume at least $2^n \det\Lambda / (n! \, \lambda_1 \cdots \lambda_n)$. The upper bound is the deep part; Minkowski's proof compares $K$ with a body built from the successive minima and uses a packing argument. Since $\lambda_1^n \le \lambda_1 \cdots \lambda_n$, the upper bound implies $\lambda_1^n \operatorname{vol}(K) \le 2^n \det \Lambda$; so if $\operatorname{vol}(K) \ge 2^n \det\Lambda$, then $\lambda_1 \le 1$ and $K$ contains a nonzero lattice point, which is Minkowski's theorem. Both bounds are sharp for $\Lambda = \Z^n$: for the cube $K = [-1, 1]^n$ all $\lambda_k = 1$ and $\operatorname{vol}(K) = 2^n$, and for the cross-polytope $\set{x : \abs{x_1} + \dots + \abs{x_n} \le 1}$ all $\lambda_k = 1$ and $\operatorname{vol}(K) = 2^n/n!$.

Going deeper

For a student

Minkowski's first theorem controls only the first minimum; the second theorem controls all of them at once. This is what one needs to find a whole basis of short vectors. It is used, for instance, in the reduction theory of quadratic forms and in bounds for units and ideals in number fields. It is also the starting point of transference theorems that relate a lattice to its dual.