Semiconvex Functions: Representations as Suprema of Smooth Functions and Extensions

We prove results on representations of semiconvex functions with an arbitrary modulus (equivalently: strongly paraconvex functions) in superreflexive Banach spaces as suprema of families of differentiable functions. Also, results on extensions of semiconvex functions are proved. Further, characterizations of semiconvex functions by uniform Fréchet subdifferentiability and (global) [α]-subdifferentiability are given. We also show that weakly convex functions in Nurminskii's sense coincide with locally semiconvex functions.