Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets

The quasidifferential calculus developed by V.F. Demyanov and A.M. Rubinov provides a complete analogon to the classical calculus of differentiation for a wide class of nonsmooth functions. Although this looks at the first glance as a generalized subgradient calculus for pairs of subdifferentials it turns out that, after a more detailed analysis, the quasidifferential calculus is a kind of Fréchet-differentiation whose gradients are elements of a suitable Minkowski–R̊adström–Hörmander space. One aim of the paper is to point out this fact. The main results in this direction are Theorem 1 and Theorem 5. Since the elements of the Minkowski–R̊adström–Hörmander space are not uniquely determined, we focus our attention in the second part of the paper to smallest possible representations of quasidifferentials, i.e. to minimal representations. Here the main results are two necessary minimality criteria, which are stated in Theorem 9 and Theorem 11.