Second-order subdifferentials and convexity of real-valued functions

In this paper we show that the positive semi-definiteness (PSD) of the Fréchet and/or Mordukhovich second-order subdifferentials can recognize the convexity of C1C1 functions. However, the PSD is insufficient for ensuring the convexity of a locally Lipschitz function in general. A complete characterization of strong convexity via the second-order subdifferentials is also given.