Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 12 (2005), No. 2, 315--329Copyright Heldermann Verlag 2005 Filling the Gap between Lower-C1 and Lower-C2 Functions Aris Daniilidis Dep. de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain arisd@mat.uab.es Jérôme Malick INRIA, Rhone-Alpes, 655 avenue de l'Europe, Montbonnot, St. Martin, 38334 Saint Ismier, France jerome.malick@inria.fr [Abstract-pdf] The classes of lower-$C^{1,\alpha}$ functions ($0<\alpha\leq 1$), that is, functions locally representable as a maximum of a compactly parametrized family of continuously differentiable functions with $\alpha$-H\"{o}lder derivative, are hereby introduced. These classes form a strictly decreasing sequence from the larger class of lower-$C^1$ towards the smaller class of lower-$C^2$ functions, and can be analogously characterized via perturbed convex inequalities or via appropriate generalized monotonicity properties of their subdifferentials. Several examples are provided and a complete classification is given. Keywords: Maximum function, lower-$C^{1,\alpha}$ function, $\alpha$-weakly convex function, $\alpha$-hypomonotone operator. MSC: 26B25; 49J52, 47H05 [ Fulltext-pdf  (406  KB)] for subscribers only.