Differentiable Selections and Castaing Representations of Multifunctions

We consider set-valued mappings defined on a linear normed space with convex closed images in Rⁿ. Our aim is to construct selections that are (Hadamard) directionally differentiable using some approximation of the multifunction. The constructions assume the existence of a cone approximation given by a certain "derivative" of the mapping. The first one makes use of the properties of Steiner points. A notion of generalized Steiner points is introduced. The second construction defines a continuous selection that passes through given points of the graph of the multifunction and is Hadamard directionally differentiable at those points, with derivatives belonging to the corresponding "derivatives" of the multifunction. Both constructions lead to a directionally differentiable Castaing representation of a multifunction possessing appropriate differentiability properties. The results are applied to obtain statements about the asymptotic behavior of measurable selections of random sets via the delta method.