Structure of polytopes associated to non-additive measures via toric ideals and Gröbner bases

Abstract

In this paper we study the geometrical structure of some polytopes appearing in the study of families of non-additive measures using toric ideals and Gröbner bases. Toric ideals and Gröbner bases are tools appearing in Computational Algebra when dealing with ideals in the ring of polynomials in several variables, and they have been applied for obtaining both the faces and a triangulation of a polytope whose vertices are integer-valued. In this paper we provide examples on which we compare these tools with other ones: order polytopes and the polytope of 2-additive measures. Finally, we derive the combinatorial structure of the subfamily of 2-additive k-ary capacities.