Global error bounds for convex inequality systems in Banach spaces

We study conditions under which a global error bound in terms of a natural residual exists for a convex inequality system. Specifically, we obtain an error bound result, which unifies many existing results assuming a Slater condition. We also derive two characterizations for a convex inequality system to possess a global error bound; one is in terms of metric regularity, and the other is in terms of an associated convex inequality system. As a consequence, we show that in R-n a global error bound holds for such a system under the assumption of the zero vector in the relative interior of the domain of an associated conjugate function along with metric regularity at every point of the feasible set defined by the system. Finally, we discuss some applications of these results to convex programs.