In this paper, we consider the dominance properties of the set of the pignistic $k$-additive belief functions. Then, given $k$, we conjecture the shape of the polytope of all the $k$-additive belief functions dominating a given belief function, starting from an analogy with the case of dominating probability measures. Under such conjecture, we compute the analytical form of the barycenter of the polytope of $k$-additive dominating belief functions, and we study the location of the pignistic $k$-additive belief functions with respect to this polytope and its barycenter.