The h vector of a pyramid

Question

I have an exercise that asks me to find the h vector of a pyramid which has a d-polytope as its base, and I have no idea what to do, so I would appreciate any help, solution or intuitive approach Thank you in advance

Answer 1

From your comment:

You know the $h$-vector for the octahedron. Now imagine a pyramid in $\mathbb{R}^4$ constructed by choosing a point outside the three dimensional subspace containing the octahedron and joining it to all the vertices of the octahedron. You should be able to compute the $f$ and $h$-vectors of that pyramid in terms of the vectors for the octahedron.

Now generalize to a pyramid $P$ in $d+1$-space whose $d$ dimensional base $B$ is given. Then compute the $f$ and $h$-vectors of $P$ in terms of those for $B$.