The Polytope of Fuzzy Measures and Its Adjacency Graph

Abstract

In this paper we deal with the problem of studying the structure of the polytope of fuzzy measure for finite referential sets. We prove that the diameter of tire polytope of fuzzy measures is 3 for referentials of 3 elements or more. We also show that the polytope is combinatorial, whence we deduce that the adjacency graph of fuzzy measures is Hamilton connected if the cardinality of the referential set is not 2. We also give some results about the facets and edges of this polytope. Finally, we treat the corresponding results for the polytope given by the convex hull of monotone boolean functions.