A Uniform Approach to Hölder Calmness of Subdifferentials

For finite-valued convex functions

defined on the

n n

-dimensional Euclidean space, we are interested in the set-valued mapping assigning to each pair

(f,x) ( f , x )

the subdifferential of

f f

at

x x

. Our approach is uniform with respect to

f 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) ( f , x )

under the assumption that the subdifferential mapping viewed as a set-valued mapping from

\mathbb{R}^{n} R n

to

\mathbb{R}^{n} R n

with

f f

fixed is calm at each point of

\{x\}\times \partial f(x) { x } × ∂ f ( x )

.