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 P (N) , generated by global sections, are non-negative if i <= 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 >= 4 can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for i <= 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 F which is genereated by H-0 F 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.