Cores of Tangent Cones and Clarke's Tangent Cone

Abstract

It is known that Clarke's tangent cone at any point of any subset of Rn is always both unique and convex. By contrast, nearly all other notions of convex tangent cone in the literature are monotone in the sense that if a convex cone K is a tangent cone at a point x0 of a set C ⊆ Rn, then K′ ⊆ K, C ⊆ C′ automatically implies that K′ is a tangent cone of the same type for C′ at x0. This carries the rider that such tangent cones are not unique and, in general, even maximal (convex) tangent cones are not unique. In this paper it is shown that for most monotone types, one can identify preferred tangent cones (for C at x0), called core tangent cones of that type, such that maximal core tangent cones are unique and convex.