The geometry of consonant belief functions: Simplicial complexes of necessity measures

In this paper we extend the geometric approach to the theory of evidence in order to include the class of necessity measures, represented on a finite domain of “frame” by consonant belief functions (b.f.s). The correspondence between chains of subsets and convex sets of b.f.s is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of “simplicial complex”. In particular we focus on the set of outer consonant approximations of a belief function, showing that for each maximal chain of subsets these approximations form a polytope. The maximal such approximation with respect to the weak inclusion relation between b.f.s is one of the vertices of this polytope, and is generated by a permutation of the elements of the frame.