  This article is devoted to the stability of error bounds (local and global)
for semi-infinite convex constraint systems in Banach spaces. We provide primal
characterizations of the stability of local and global error bounds when
systems are subject to small perturbations. These characterizations are given
in terms of the directional derivatives of the functions that enter into the
definition of these systems. It is proved that the stability of error bounds is
essentially equivalent to verifying that the optimal values of several minimax
problems, defined in terms of the directional derivatives of the functions
defining these systems, are outside of some neighborhood of zero. Moreover,
such stability only requires that all component functions entering the system
have the same linear perturbation. When these stability results are applied to
the sensitivity analysis of Hoffman's constants for semi-infinite linear
systems, primal criteria for Hoffman's constants to be uniformly bounded under
perturbations of the problem data are obtained.
