The necessary and sufficient conditions for representing Lipschitz multivariable function as a difference of two convex functions

Abstract

In the article the necessary and sufficient conditions for a representation of Lipschitz function of more than two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome of this algorithm is a sequence of pairs of convex functions that converge uniformly to a pair of convex functions if the conditions of the formulated theorems are satisfied. A geometric interpretation is also given. In finally, the author considered a class of curves that have the projections on, so called, coordinate planes that bound convex compact sets on such planes. This class of curves is used for formulation the necessary and sufficient conditions for solving this problem.