The geometry of consonant belief functions: simplicial complexes of possibility measures

In this paper we extend the geometric approach to the theory of evidence in order to include other important finite fuzzy measures. In particular we describe the geometric counterparts of the class of possibility measures represented by consonant belief functions. The correspondence between chains of subsets and convex sets of consonant functions 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. Exploiting the duality of the associated norms we consider the consonant approximation problem in a simple case study, compare the geometry of the solutions with that of inner and outer approximations, and formulate conjectures on their general behavior. We also propose an optimization criterion based on Dempster's rule of combination. Abstract