Lipschitz functions with minimal Clarke subdifferential mappings

In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdifferential mapping of a real-valued locally Lipschitz function is a minimal weak cusco. We then use this characterisation to deduce some new results concerning Lipschitz functions with minimal subdifferential mappings. AMS Maths Subject Class (1991): 49J52 In the papers [7] and [1], the authors investigate a class of locally Lipschitz functions that possess `generic' differentiability properties which are similar to those enjoyed by convex functions. This paper, continues this investigation. Let (X; jj \Delta jj) be a Banach space. We will call a Borel subset N ` X a Haar-null set if there exists a (not necessarily unique) Radon probability measure p on X such that p(x +N) = 0 for each x 2 X. (In this case, we call p a test-measure for N .) More generally, we say that a subset N ` X is a Haar-null set if it is contained in a Borel Haar-null set. Below, we list some of the properties enjoyed ..