Geometry of the set of dominating k-additive belief functions

In this paper we introduce a novel, simpler form of the polytope of inner Bayesian approximations of a belief function, or “consistent probabilities”. We prove that the set of vertices of this polytope is generated by all possible permutations of elements of the domain, mirroring a similar behavior of outer consonant approximations. An intriguing connection with the behavior of maximal outer consonant approximations is highlighted, and the notion of inner (outer) approximation of a credal set in terms of lower probabilities proposed. Finally, we generalize the main result to the case of k-additive belief functions, belief functions whose focal elements have size at most k. We prove that the set of such objects dominating a given belief function is also a polytope whose vertices are generated by permutations of focal elements of size at most k.