The discrete mathematics of consistent belief functions
In this paper we extend the geometric approach to belief functions to the case of consistent belief functions, i.e. belief functions whose plausibility function is a possibility measure, as a step towards a unifled geometric picture of a wider class of fuzzy measures. We prove that, analogously to consonant belief functions, they form a simplicial complex (a structured collection of simplices) whose maximal simplices are congruent. We also show that consistency is strictly related to the issue of combinability, a fact re∞ected in the similarity between the consistent complex and the complex of singular belief functions, i.e. belief functions whose core is a proper subset of their domain. Finally we draw inspiration from the approximation problem to illustrate a convex decomposition of belief functions in terms of some \consistent coordinates" on the complex which is closely related to the pignistic function, an example of how the language of discrete mathematics can help us to draw connections between belief, probability and possibility.