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)\,-\,H\"{o}lder calmness of our mapping at a nominal pair $(f,x)$ under the assumption that the subdifferential mapping viewed as a set-valued mapping from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ with $f$ fixed is calm at each point of $\{x\}\times \partial f(x)$.