Metric subregularity of the convex subdifferential in Banach spaces

In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated by some recent results in [16] where the authors extend to Banach spaces the characterization of the strong regularity, we extend as well the characterizations for the metric subregularity and the strong subregularity given in [2] to Banach spaces. We also notice that at least one implication in these characterizations remains valid for the limiting subdifferential without assuming convexity of the function in Asplund spaces. Additionally, we show some direct implications of the characterizations for the convergence of the proximal point algorithm, and we provide some characterizations of the metric subregularity and calmness properties of solution maps to parametric generalized equations