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More than first-order developments of convex functions: primal-dual relations. (English) Zbl 0868.49014
Summary: 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.

##### MSC:
 49J52 Nonsmooth analysis 26B25 Convexity of real functions of several variables, generalizations 52A41 Convex functions and convex programs in convex geometry
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