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 $\Sigma$ of $n$-dimensional simplices with vertices in $E$ and the system $\Gamma$ of chambers. The incidence matrix $A= \parallel a_{\sigma, \gamma}\parallel$, $\sigma \in \Sigma$, $\gamma \in \Gamma$, induces the notion of linear independence among simplices (and among chambers). We present an algorithm of construction of bases of simplices (and bases of chambers). For the case $n=2$ such an algorithm was described in the author's paper {\em Combinatorial bases in systems of simplices and chambers} (Discrete Mathematics 157 (1996) 15--37). However, the case of $n$-dimensional space required a different technique. It is also proved that the constructed bases of simplices are geometrical.