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Journal of Convex Analysis 23 (2016), No. 3, 719--732
Copyright Heldermann Verlag 2016

Weak Convexity of Functions and the Infimal Convolution

Grigorii E. Ivanov
Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgoprudny -- Moscow region, Russia 141700

In terms of sup-inf convolutions we characterize the class CWC(γ) of convolutionally weakly convex functions, that is a special class of Φ-convex functions, generated by the function γ . Motivated by Moreau-Yosida and Lasry-Lions regularizations, we consider a subclass of CWC(γ), namely, the class RCWC(γ) of regularly convolutionally weakly convex functions. We show that under reasonable assumptions on γ the class RCWC(γ) is the same as the class WC(γ), introduced from other considerations.

Keywords: Weak convexity, infimal convolution, Moreau-Yosida regularization, Lasry-Lions regularizations.

MSC: 41A50, 41A65, 52A21

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