On extreme points of p-boxes and belief functions

Within imprecise probability theory, the extreme points of convex probability sets have an important practical role (to perform inference on graphical models, to compute expectation bounds, …). This is especially true for sets presenting specific features that make them easy to manipulate in applications. This easiness is the reason why extreme points of such models (probability intervals, possibility distributions, …) have been well studied. Yet, imprecise cumulative distributions (a.k.a. p-boxes) constitute an important exception, as the characterization of their extreme points remain to be studied. This is what we do in this paper, where we characterize the maximal number of extreme points of a p-box, give a family of p-boxes that attains this number and show an algorithm that allows to compute the extreme points of a given p-box. To achieve all this, we also provide what we think to be a new characterization of extreme points of a belief function.