Combinatorial Structure of the Polytope of 2-Additive Measures

In this article, we study the polytope of 2-additive measures, an important subpolytope of the polytope of fuzzy measures. For this polytope, we obtain its combinatorial structure, namely the adjacency structure and the structure of two-dimensional faces, three-dimensional faces, and so on. Based on this information, we build a triangulation of this polytope satisfying that all simplices in the triangulation have the same volume. As a consequence, this allows a very simple and appealing way to generate points in a random way in this polytope, an interesting problem arising in the practical identification of 2-additive measures. Finally, we also derive the volume and the centroid of this polytope.