An infinite 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 infinite 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 cone of Clarke and the B-tangent cone or the cone of Michel and Penot, warrant further study. The relationship between these new cones and the differentiability of functions is also considered.