About $f$-vectors of pyramidal triangulations of point configurations

A triangulation of a point configuration is called pyramidal if all its simplexes have a common vertex. Some inequalities for the components of the $f$-vectors of pyramidal triangulations were established. Moreover, for each $d>3$ there was constructed a $d$-dimensional polytope with its triangulation $T(d)$ such that the $f$-vector of $T(d)$ is not realizable as the $f$-vector of a pyramidal triangulation. Bibl. 13.