urvey on some classical and new inequalities involving Minkowski's
successive minima $\lambda_i(K)$ of a ($o$-symmetric) convex body
$K\subset\RR^n$, where $\lambda_i(K)$ is the smallest positive number
$\lambda$ such that $\lambda\,K$ contains at least $i$ linearly
independent lattice points of $\ZZ^n$. Minkowski proved bounds for the
volume of $K$ in terms of the successive minima, and here we want to
discuss possible extensions/generalizations of these inequalities
when the volume is replaced by
the lattice point enumerator or when the successive minma are subject
to certain  restrictions.