A study of the $k$-additive core of capacities through achievable families

抄録

We investigate in this paper about the set of $k$-additive capacities dominating a given capacity, which we call the $k$-additive core. We study its structure through achievable families, which play the role of maximal chains in the classical case ($k=1$), and show that associated capacities are element (possibly a vertex) of the $k$-additive core when the capacity is $(k+1)$-monotone. The problem of finding all vertices of the $k$-additive core is still an open question.