  It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on
$\PP^N$, generated by global sections, are non-negative if $i\leq n$ and vanish
otherwise. This paper deals with the following question: does the above result
hold for the wider class of reflexive sheaves? We show that the Chern numbers
$c_i$ with $i\geq 4$ can be arbitrarily negative for reflexive sheaves of any
rank; on the contrary for $i\leq 3$ we show positivity of the $c_i$ with weaker
hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every
reflexive sheaf $\FF$ which is generated by $H^0\FF$ on some non-empty open
subset and completely classify sheaves for which either of them reach the
minimum allowed, or some value close to it.
