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Choquet theorem and nonsmooth analysis. (English) Zbl 0679.49015
It is shown that a classical theorem of G. Choquet [Ann. Univ. Grenoble, Sect. Sci. Math. Phys. II. Sér. 23, 57-112 (1948; Zbl 0031.28101)] (stating the generic equality between the contingent and the paratingent cones to a subset of a separable Banach space) has several applications to nonsmooth analysis.
In particular, it is proved that if an open subset of a separable Banach space is generically tangentially regular (i.e. regular in the sense of F. H. Clarke [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)]) on its boundary, then its contingent cone is generically equal to a closed halfspace.
Next, it is shown that for a continuous function on an open subset of a separable Banach space, the subdifferential regularity, the regular Gâteaux differentiability and the strict differentiability in the full limit sense are generically equivalent.
Reviewer: M.Studniarski

##### MSC:
 49J52 Nonsmooth analysis 46G05 Derivatives of functions in infinite-dimensional spaces 49J50 Fréchet and Gateaux differentiability in optimization