METRICALLY DIFFERENTIABLE SET-VALUED FUNCTIONS AND THEIR LOCAL LINEAR APPROXIMATES

A new notion of metric differentiability at a point of set-valued func­tions with general compact sets in Rⁿ as values is introduced. Extending the clas­sical approach, we use right and left limits of set-valued metric divided differences of first order. A local metric linear approximant of a metrically differentiable set-valued function at a point is defined and studied. This local approximant may be regarded as a special realization of the set-valued Euler approximants of M. S. Nikolskii and the directives of Z. Artstein. Error estimates for the local metric linear approximant are obtained. In particular, a second order approxi­mation is derived for a class of strongly metrically differentiable set-valued maps.