In this paper we solve the problem of approximating a belief measure with a necessity measure or "consonant belief function" by minimizing appropriate distances from the consonant complex in the space of all belief functions. Partial approximations are first sought in each simplicial component of the consonant complex, while global solutions are obtained from the set of partial ones. The L_1, L_2 and L∞ consonant approximations in the belief space are here computed, discussed and interpreted as generalizations of the maximal outer consonant approximation. Results are also compared to other classical approximations in a ternary example.
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