Positivity of Chern Classes for Reflexive Sheaves on P^N

It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on ${\text{PP}}^N$, generated by global sections, are non-negative if $i\le n$ 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 $i\ge 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\le 3$ 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 $\text{FF}$ which is generated by $H^0\text{FF}$ 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.