More Than First-Order Developments of Convex Functions: Primal-Dual Relations

The subject of this paper concerns the remainder term in the first-order development of a (finite-valued) convex function. We study functions for which this term is comparable to a squared norm and we relate it to the corresponding remainder term of the conjugate function. We show that a convex function satisfies a quadratic growth condition if and only if its subdifferential satisfies a linear growth condition. Finally, we define a new concept of “tangential regularization”, involving a local decomposition of $\mathbb{R}^N$, along the subspace where the function is “smooth” and the subspace parallel to the subdifferential.