Combinatorial bases in systems of simplices and chambers

Abstract

We consider a finite set E of points in the n-dimensional affine space and two sets of objects that are generated by the set E: the system Σ of n-dimensional simplices and the system Γ of chambers. The relation ( A; Σ, Γ) introduced by the incidence matrix M = | a σ,γ| defines the notion of linear independence and the rank of the system of simplices and of the system of chambers. We introduce the notion of a combinatorial basis. Combinatorial bases of chambers can be described in terms of a game. We describe the algorithm of decomposition of a convex polytope into shells. In the case of the affine plane, using the game and the algorithm we construct a combinatorial basis B of chambers. Using the algorithm, we also construct a basis B′ of simplices that together with the basis B of chambers form a ‘triangular pair’.