Generalized Lipschitz functions

The aim of this paper is to establish a compactness result on some function sets. The main idea is very simple : it suffices to change the axis in order to transform a family of nondecreasing functions in Lipschitz ones and then to apply Ascoli's theorem. As we will see, this simple geometrical approach can be extended to a wider class of functions. The paper is organized as follows. In the next section we shall define the concept of Q-Lipschitz functions, where Q is a convex cone and we shall construct a particular topology on this set. In Section 2, we shall establish our compactness result and we shall explore some properties of the considered topology. In Section 3, we shall extend the previous result to a more general class of functions and in Section 4 we shall present some applications of our result.