On global properties of lower semicontinuous quadratically minorized functions

We use the framework of a type of abstract convexity ($\Phi_{lsc}$-convexity) to investigate properties of lower semicontinuous quadratically minorized functions in Hilbert spaces. A new result, which states that, for every local $\Phi_{lsc}$-subgradient there exists a global one is proved and plays a crucial role in our considerations. We deliver conditions for abstract subdifferentiability ($\Phi_{lsc}$-subdifferentiability) of locally $C^{1,1}$ functions, twice continuously differentiable functions, prox-regular functions and paraconvex functions. As an application we establish a new sufficient and necessary condition for minimax equality for $\Phi_{lsc}$-convex functions. This new condition is expressed in therms of $\Phi_{lsc}$-subdifferential.