Rotund Norms, Clarke Subdifferentials and Extensions of Lipschitz Functions

. We show that a certain condition regarding the separation of points by Lipschitz functions is useful in extending a given Lipschitz function from a subspace of a Banach space to the whole space so that the extension function has maximal Clarke subdierential. We then establish connections between this separation property and the rotundity properties of the Banach space. AMS Classication: 49J52, 46B20. 1. Introduction. The Clarke derivative of a locally Lipschitz function is dened by f Æ (x; v) := lim sup t # 0 y ! x f(y + tv) f(y) t ; and the Clarke subdierential is dened by @ c f(x) = f 2 X : (v) f Æ (x; v) for all v 2 Xg. This subdierential has been widely used as a powerful tool in nonsmooth analysis with applications in diverse areas of optimization; see [4] for more information. Recently, substantial progress has been made on understanding the limitations of the Clarke derivative, in particular, see [2,3] and the references therein. Among other things, it ..