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 $\mathbb{R}^n$)?