An In nite Class of Convex Tangent Cones

Since the early 1970's there have been many papers devoted to tangent cones and their applications to optimization. Much of the debate over which tangent cone is \best" has centered on the properties of Clarke's tangent cone and whether other cones have these properties. In this paper it is shown that there are an innnite number of tangent cones with some of the nicest properties of Clarke's cone. These properties are convexity, multiple characterizations, and proximal normal formulas. The nature of these cones indicates that the two extremes of this family of cones, the cones of Clarke and B{tangent cone or the cone of Michel and Penot, warrant further study. The relationship between these new cones and the diierentiability of functions is also considered.