Let $\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n$ be $n$ linearly independent vectors in an $n$-dimensional lattice $\Lambda$ in $\mathbf{R}^n$ and let $\mathbf{v}^*_1 ,\mathbf{v}^*_2, ..., \mathbf{v}^*_n$ denote the Gram-Schmidt orthogonalization of $\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n$.

I'm interested in the following quantity:

$$f_n(\Lambda) =\min_{{\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n \in \Lambda}} \{\max_i \Vert\mathbf{v}^*_i\Vert \}$$

Notice that — in a definition similar to Minkowski's successive minima — $f_n(\Lambda)$ is the smallest radius of a ball in $\mathbf{R}^n$ that contains $n$ linearly independent Gram–Schmidt vectors generated by $\Lambda$.

In particular we have that $$(1)\;\;f_n(\Lambda)\leq \lambda_n(\Lambda),$$ where $\lambda_n(\Lambda)$ is the $n$-th successive minimum of $\Lambda$, and $$\;\;\;\;\; \ (2)\;\;\det(\Lambda)^{{1}/{n}} \leq f_n(\Lambda).$$

So I come to my question: Is it possible to bound (1) or (2) in the opposite direction to make a statement that qualitatively reads: $f_n(\Lambda)$ is not much larger than $\det(\Lambda)^{{1}/{n}}$ and/or not much smaller than $\lambda_n(\Lambda)$?

For example, say something like:

$$ f_n(\Lambda)\leq c_1 \det(\Lambda)^{{1}/{n}} \;\;\;\mbox{ and/or }\;\;\; f_n(\Lambda)\geq \frac{1}{c_2}\lambda_n(\Lambda) $$

for some quantities $c_1,c_2$ that are "not too large" ?