In this paper the geometric structure of the space $\mathcal{S}_{\Theta}$ of the belief functions defined over a discrete set $\Theta$ (belief space) is analyzed. Using the Moebius inversion lemma we prove the recursive bundle structure of the belief space and show how an arbitrary belief function can be uniquely represented as a convex combination of certain elements of the fibers, giving $\mathcal{S}$ the form of a simplex. The commutativity of orthogonal sum and convex closure operator is proved and used to depict the geometric structure of conditional subspaces, i.e. sets of belief functions conditioned by a given function s. Future applications of these geometric methods to classical problems like probabilistic approximation and canonical decomposition are outlined.
Keywords. Theory of evidence, belief space, fiber bundle, convex decomposition, commutativity, conditional subspace.
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Authors addresses:
Fabio Cuzzolin
Dipartimento di Elettronica e Informatica
Via Ognissanti 72
35131 Padova
Italy
Ruggero Frezza
Dipartimento di Elettronica e Informatica
Via Ognissanti 72
35131 Padova
Italy
E-mail addresses:
| Fabio Cuzzolin | cuzzolin@dei.unipd.it |
| Ruggero Frezza | frezza@dei.unipd.it |