Weak Convexity of Functions and the Infimal Convolution

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.