Combinatorial Structure of the Polytope of 2-Additive Measures

Abstract

In this paper we study the polytope of 2-additive measures, an important subpolytope of theIn this paper 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 2-dimensional faces, 3-dimensional faces, and so on. Basing 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 problema arising in the practical identi_cation of 2-additive measures. Finally, we also derive the volume, the centroid, and some properties concerning the adjacency graph of this polytope. polytope of fuzzy measures. For this polytope, we obtain its combinatorial structure, namely the adjacency structure and the structure of 2-dimensional faces, 3-dimensional faces, and so on. Basing 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 problema arising in the practical identi_cation of 2-additive measures. Finally, we also derive the volume, the centroid, and some properties concerning the adjacency graph of this polytope.