I have the following theorem that I would like to prove:
Let $\Lambda$ be an $m$-dimensional Lattice of $\mathbb{R}^n$ then $\Lambda$ has a basis consisting of $b_1,...,b_m$ with $||b_j||= \lambda_j(\Lambda)$ for $1 \leq j \leq min(m,4)$ where $\lambda_j(\Lambda)$ denotes the $j$-th successive minima.
For smaller dimensions, however, I can only find examples that confirm the theorem. How would one prove something like this in general?
I think this is an interesting question that may be of interest to the community here.