Proximal Mappings

Abstract

We study differentiability properties and subdifferentiability properties of the Baire approximate and of the Moreau–Yosida approximate of a nonconvex function on a Banach space. These properties are intimately linked with exactness of the infimal convolution defining the approximation. When applied to indicator functions of possibly nonconvex subsets, our results yield existence of best approximations under subdifferentiability assumptions on the distance function and suitable smoothness assumptions on the space.