A Uniform Approach to Hölder Calmness of Subdifferentials

For finite-valued convex functions f defined on the n-dimensional Euclidean space, we are interested in the set-valued mapping assigning to each pair (f, x) the subdifferential of f at x. Our approach is uniform with respect to f in the sense that it involves pairs of functions close enough to each other, but not necessarily around a nominal function. More precisely, we provide lower and upper estimates, in terms of Hausdorff excesses, of the subdifferential of one of such functions at a nominal point in terms of the subdifferential of nearby functions in a ball centered in such a point. In particular, we obtain the (1/2) - Hölder calmness of our mapping at a nominal pair (f, x) under the assumption that the subdifferential mapping viewed as a set-valued mapping from Rn to Rn with f fixed is calm at each point of {x} X ∂f(x).This research has been partially supported by Grant MTM2014-59179-C2-(1-2)-P from MINECO, Spain, and FEDER “Una manera de hacer Europa”, European Union. The third author was partially supported by the Australian Research Council, Project DP160100854