On Locally Polyhedral Convex Functions

A specific property of convex functions, which is called the diff-max property, plays an important role in some aspects of optimization. This paper shows that in a finite dimensional space a closed proper convex function has this property if and only if it is locally polyhedral. A preliminary study of closed locally polyhedral convex sets is provided and a survey of some applications of the diff-max property in optimization is given.