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 $\mathbb{R}^n$)? 



Minkowski successive minima inequality for a lattice base?
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](http://www.ams.org/journals/tran/1940-048-01/S0002-9947-1940-0002345-2/).


