In Chapter [IX.1] of Siegel's *Lectures on the Geometry of Numbers* it is shown that if we have $n$ linear forms $y_{j}=\sum_{k=1}^{n}{a_{jk}x_{k}},\quad j=1,\ldots,n$, with the coefficient matrix $(a_{jk})$ being non-singular and having determinant $D$, then

$$ \min_{(x_{i})_{i}^{n} \in \mathbb{Z}^{n}-\{0\}}|y_{1}\ldots y_{n}| \leq\frac{n!|D|}{n^{n}}. $$

This bound is not, however, best possible. Siegel goes on to prove that for $n=2$ the best possible upper bound on $|y_{1}y_{2}|$ is $\sqrt{\frac{1}{5}}|D|$.

My question is: let $|y_{1}\ldots y_{n}| \leq c_{n}|D|$ be the best possible upper bound, what is known about $c_{n}$?

Siegel's result is $c_{2}=\frac{1}{\sqrt{5}}$ and he also credits Davenport with showing that $c_{3} \leq \frac{1}{7}$.

What else has been found since?