Convexity and Openness with Linear Rate

This paper presents conditions for openness with linear rate or, equivalently, for metric regularity. of continuous mappings that possess certain convexity properties. Convex continuous functions on a Banach space are proved to be open with linear rate around each point that is not a minimum point. For continuous mappings that are convex with respect to a normal cone in a finite dimensional Banach space as image space, a sufficient condition for openness with linear rate is given. Special cases are treated: For Fr´echet-differentiable cone]convex mappings, the surjectivity of the derivative is proved to be equivalent to openness with linear rate. Finitely generated cones lead to a sufficient condition for openness with linear rate that simplifies practical use. A tangency formula of Lyusternik-type is set up for mappings that are open with linear rate, and is applied to cone]convex mappings.