On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions

TLDR: In this paper, the differential calculus for convex, compact-valued multifunctions developed by Tyurin, Banks and Jacobs is used to give an equivalent description in terms of multifunctions of the class of functions which can be represented as the difference of two globally Lipschitzian convex functions.

Abstract: The differential calculus for convex, compact-valued multifunctions developed by Tyurin, Banks and Jacobs is used to give an equivalent description in terms of multifunctions of the class of functions which can be represented as the difference of two globally Lipschitzian convex functions. This approach is also used to develop a means of representing piecewise-linear continuous functions as the difference of two piecewise-linear convex functions in finite dimensions. This leads directly to a Minkowski duality theorem for piecewise-linear positively homogeneous continuous functions and equivalence classes of convex compact sets produced by convex compact polyhedrons: every piecewise-linear positively homogeneous continuous function may be uniquely characterized by its quasidifferential (as defined by Demyanov and Rubinov) at zero.