Geometry of the set of dominating k-additive belief functions Fabio Cuzzolin, Thomas Burger and Alessandro Antonucci Department of Computing and Communication Technologies, Oxford Brookes University, Oxford, UK. Abstract 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. Key words: Theory of evidence, consistent probabilities, inner Bayesian approximations, outer consonant approximations, k-additive belief functions, information ordering, permutation. PACS: 1 Introduction Uncertainty theory is a composite field in which different but related ap- proaches compete to gain a wider audience in engineering [1] and business [2] applications. Belief [3], probability, and possibility [4] measures can all be adopted to represent uncertainty, although some of them may be more fit to Email address: Fabio.Cuzzolin@brookes.ac.uk (Fabio Cuzzolin, Thomas Burger and Alessandro Antonucci). URL: http://cms.brookes.ac.uk/staff/FabioCuzzolin/ (Fabio Cuzzolin, Thomas Burger and Alessandro Antonucci). Preprint submitted to Elsevier Science 8 July 2011