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Journal of Convex Analysis 12 (2005), No. 2, 315--329
Copyright 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

Jérôme Malick
INRIA, Rhone-Alpes, 655 avenue de l'Europe, Montbonnot, St. Martin, 38334 Saint Ismier, France


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

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