Lipschitz Functions with Minimal Clarke Subdiierential Mappings
نویسندگان
چکیده
In this paper we characterise, in terms of the upper Dini derivative, when the Clarke subdiierential mapping of a real-valued locally Lipschitz function is a minimal weak cusco. We then use this characterisation to deduce some new results concerning Lips-chitz functions with minimal subdiierential mappings. In the papers 7] and 1], the authors investigate a class of locally Lipschitz functions that possess`generic' diierentiability properties which are similar to those enjoyed by convex functions. This paper, continues this investigation. Let (X; jj 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 by Haar-null sets.
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