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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.

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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