Complexes of Outer Consonant Approximations

In this paper we discuss the problem of approximating a belief function (b.f.) with a necessity measure or “consonant belief function” (co.b.f.) from a geometric point of view. We focus in particular on outer consonant approximations, i.e. co.b.f.s less committed than the original b.f. in terms of degrees of belief. We show that for each maximal chain of focal elements the set of outer consonant approximation is a polytope. We describe the vertices of such polytope, and characterize the geometry of maximal outer approximations.