One of the uses of the derivative of a nonlinear function F is to define first-order approximations of F by special functions, namely affine functions. The present aim is to extend this concept of first order approximations to the case when F is multi-valued; how should the concept of affinity be generalized for such functions? The authors first show why a mere copy of the single-valued case is hardly appropriate; a more sophisticated definition is needed for affinity. Restricting the present study to convex compact valued F, they consider various possibilities. The authors conclude by exhibiting a class of multi-valued mappings, which they call eclipsing, and which seem to impose themselves as natural generalizations of affine mappings. They correspond to linearizing the support function of F, which thus offers to be the most reasonable thing to do.
