$L_{p}$ Consonant Approximations of Belief Functions

In this paper, we solve the problem of approximating a belief measure with a necessity measure or “consonant belief function” in a geometric framework. Consonant belief functions form a simplicial complex in both the space of all belief functions and the space of all mass vectors: Partial approximations are first sought in each component of the complex, while global solutions are selected among them. As a first step in this line of study, we seek here approximations that minimize L_p norms. Approximations in the mass space can be interpreted in terms of mass redistribution, while approximations in the belief space generalize the maximal outer consonant approximation. We compare them with each other and with other classical approximations and illustrate them with the help of a running example.