DENSELY DEFINED SELECTIONS OF MULTIVALUED MAPPINGS

Rather general suficient conditions are found for a given multivalued mapping F: X --> Y to possess an upper semicontinuous and compact-valu ed selection G which is defined on a dense G(delta)-subset of the doma in of F. The case when the selection G is single-valued (and continuou s) is also investigated. The results are applied to prove some known a s well as new results concerning generic differentiability of convex f unctions, Lavrentieff type theorem, generic well-posedness of optimiza tion problems and generic non-multivaluedness of metric projections an d antiprojections.