Bases in Systems of Simplices and Chambers

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 $\mathrm{\Sigma }$ of $n$-dimensional simplices with vertices in $E$ and the system $\mathrm{\Gamma }$ of chambers. The incidence matrix $A=\parallel a_{\sigma ,\gamma }\parallel$, $\sigma \in \mathrm{\Sigma }$, $\gamma \in \mathrm{\Gamma }$, 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.