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Let $\Lambda$ be a lattice of $\mathbb{R}^n$, and $\lambda_i$ be the radius of the smallest ball containing $i$ linearly independent lattice vectors.

The Minkowski successive minima inequality says that: $ (\prod_{i=1}^n \lambda_i)^{1/n} \leq \sqrt{n} (\det \Lambda)^{1/n} $

Can we also have an upper bound if in addition, we ask these independent vectors to be a basis of the lattice (and not only a basis of $\mathbb{R}^n$)?

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Yes, a version of Minkowski's successive minima studied by Mahler and Weyl consists in letting $\lambda_i'$ to be the radius of the smallest ball containing $i$ linearly independent lattice vectors that can furthermore be completed to a basis of the lattice. The inequality of Minkowski holds with a somewhat worse constant. This can be found in Lekkerkerker's book or in Wey'ls paper.

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Since any lattice has a basis of length at most $\sqrt n \lambda_n$, there is always a basis of length at most $n (\det\Lambda)^{1/n}$.

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