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The geometric approach to the theory of evidence can be extended to include the class of necessity measures, represented on a finite domain or
'frame' by consonant belief functions (b.f.s). The correspondence between chains of subsets and convex sets of b.f.s 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'.
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We can solve the problem of approximating a belief measure with a possibility measure or "consonant belief function" in our geometric framework.
As possibility measures are strictly related to the L1 norm it makes sense to look for approximations which minimize Lp norms. Partial approximations are first sought in each simplicial component of
the consonant complex, while global solutions are obtained from the set of such partial ones. The obtained Lp consonant approximations are discussed
and their interpretation in terms of degrees of belief provided.
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The class of consistent belief functions can be introduced as the counterpart of consistent knowledge bases in classical logic.
As consistency can be desirable in decision
making, the problem of transforming an arbitrary belief function into a consistent one arise. This can be posed
in a geometric setup, as consistent belief functions live on a structured collection of simplices called
simplicial complex. Indeed, each belief function naturally decomposes into consistent components on such a complex, in a fashion which recalls
the pignistic transform.
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Consistent belief functions represent collections of coherent or non-contradictory pieces of evidence. As most operators used to update or elicit
evidence do not preserve consistency, the use of consistent transformations in a reasoning process to guarantee coherence can be desirable. These
are in turn linked to the problem of approximating an arbitrary belief function with a consistent one.
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