The complexity of subdifferentiation and its inverse on convex functions in Banach spaces

Let E be a separable Banach space with separable dual. We show that the operation of subdifferentiation and the inverse operation are Borel from the convex functions on E into the monotone operators on E (subspace of the closed sets of E x E*) for the Effros-Borel structures. We also prove that the set of derivatives of differentiable convex functions is coanalytic non-Borel, by using the already known fact that the set of differentiable convex functions is itself coanalytic non-Borel, as proved in Bossard et al. (J. Funct. Anal. 140 (1) (1996) 142). At last, we give a new proof of this latter fact, for reflexive E's, by giving a coanalytic rank on those sets and constructing functions of "high ranks". This approach, based on an ordinal rank which follows from a construction of trees, is quite different - not so general but actually more constructive - from the previous results of this kind, in Bossard et al. (J. Funct. Anal. 140 (1) (1996) 142) and Godefroy et al. (Proc. Mons Conf. Anal. Logic, Ann. Pure Appl. Logic, in press), based on reductions of arbitrary coanalytic or difference of analytic sets to the studied sets. (C) 2002 Published by Elsevier Science B.V.