Boundary behavior of supply: A continuity property of the maximizing correspondence

The normal cone correspondence, which assigns to each point of a convex set the cone of normal vectors to the set at this point, is well known not to be lower hemi-continuous. Sufficient 'conditions for a weaker approximability condition of some points in the graph of this correspondence are given. The result is applied to study the boundary behavior of supply functions of producers who maximize profit on a convex set.