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

✍ Scribed by Pierre Casevitz

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
200 KB
Volume
118
Category
Article
ISSN
0168-0072

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✦ Synopsis

Let E be a separable Banach space with separable dual. We show that the operation of subdi erentiation 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 Γ— E * ) for the E ros-Borel structures.

We also prove that the set of derivatives of di erentiable convex functions is coanalytic non-Borel, by using the already known fact that the set of di erentiable 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 re exive 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 di erent -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 di erence of analytic sets to the studied sets.

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