Let P be a d dimensional (convex) polytope, and Q a face of P. Let Q' be a translation of Q which is outside the affine hull of P (i.e., Q' contains a vertex not in the affine hull of P). Given the face lattices of P and Q, I am seeking results (if any, and including special cases) (aside from the two special cases given below) about the face lattice of conv(P $\bigcup$ Q'). Thanks!
Two special cases are:
If Q is a vertex, then conv(P $\bigcup$ Q') is a pyramid over P.
If Q = P, then conv(P $\bigcup$ Q') is a prism over P (i.e., a rectangular product of P and a line segment).