Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The **$i$-th successive minima** of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of radius of the balls containing $i$-linearly independent vectors in $L$. 

For example, the first successive minima is the shortest nonzero vector in $L$ and the second successive minima is the second shortest vector in $L$ that is linearly independent of the first one.

I wonder if there are estimates/exact values for the following quantities for $1\le i \le d$:

$$A_i := \inf \{\lambda_i(L):L\in \Lambda\},$$

$$B_i := \sup \{\lambda_i(L):L\in \Lambda\}.$$

I know from [Minkowski's second convex body theorem][1] that two of them, namely $B_1 < \infty$ and $A_d >0$. In addition, $A_1=0$ via the following example
$ \begin{bmatrix}
e^{\frac{d-1}{d}t}I_{d-1} & 0 \\
0 & e^{-\frac{t}{d}} 
\end{bmatrix}  \mathbb Z^{d}$ as $t\to \infty$.

For these $2d$ quantities, I don't have to know their specific values, but I would like to know if they are finite/infinite/zero/nonzero. Are there ways (powerful theorems) to see they values? Also I haven't found this discussed in any books in geometry of numbers or Diophantine approximations.


  [1]: https://en.wikipedia.org/wiki/Minkowski%27s_second_theorem