Abstract

In terms of sup-inf convolutions we characterize the class CWC(γ)\mathcal{CWC}(\gamma) 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)\mathcal{CWC}(\gamma), namely, the class RCWC(γ)\mathcal{RCWC}(\gamma) of regularly convolutionally weakly convex functions. We show that under reasonable assumptions on γ the class RCWC(γ)\mathcal{RCWC}(\gamma) is the same as the class WC(γ)\mathcal{WC}(\gamma), introduced from other considerations.

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Contact details are reproduced from the original publication and may be historical.

Grigorii E. Ivanov

Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgoprudny – Moscow region, Russia 141700

givanov@mail.mipt.ru

Keywords

  • Weak convexity
  • infimal convolution
  • Moreau-Yosida regularization
  • Lasry-Lions regularizations

Mathematics Subject Classification

41A50, 41A65, 52A21

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Suggested citation

G. E. Ivanov. “Weak Convexity of Functions and the Infimal Convolution.” Journal of Convex Analysis 23 (2016), No. 3, 719–732. https://doi.org/10.68381/jca23028

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Published by Heldermann Verlag, 2016. Rights now held by Banach Press LLC.