Let $L\le \mathbb R^n$ be a lattice and $g\in GL(n,\mathbb R)$. Suppose $g$ is fixed but $L$ is allowed to vary.
Let $\lambda_i(L)$ be the $i$-th minimum of the lattice $L$ for $1\le i \le n$. I wonder if there are lower and upper bounds of $\lambda_i(gL)/\lambda_i(L)$ that are independent of the choice of $L$ and $1\le i \le n$ (but could depend on $g$ and $n$).
I am inclined to believe that this is false and trying to come up with a lattice and $g$ such that the ratio above could be very small. There is an upper bound when $i=1$ and lower bound when $i=n$. See the link below.
What I asked is different from the following thread, where we compared different minima:
How does the action of $GL(n,\mathbb R)$ affect the $k$-th minimum of a lattice?