The U-Lagrangian of a prox-regular function

When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function f and a point x, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian can be defined and has nice smooth properties and V, the orthogonal complement subspace of U. In this paper we generalize the definition of UV-decomposition and U-Lagrangian to the context of nonconvex functions, specifically that of a prox-regular function.