Weak Convexity of Functions and the Infimal Convolution

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