Use of strict twice epidifferentiability

Identify theoretical results that make use of the strict twice epidifferentiability of the convex functional $G$, in which the recovery sequences for the weak-$\star$ second subderivative additionally satisfy norm convergence.

Background

The paper distinguishes weak-$\star$ twice epidifferentiability from the stronger property of strict twice epidifferentiability, where the recovery sequences also satisfy $\|h_k\|_X \to \|h\|_X$. The cited computations establish strict twice epidifferentiability for certain functionals, but the paper does not identify any theoretical results that exploit this stronger property. Determining such results would clarify whether strict twice epidifferentiability yields consequences beyond those obtained from ordinary weak-$\star$ twice epidifferentiability.