Are there face lattice results for the following class of polytope constructions? 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).
 A: This is more of a description of a possible approach to this question, rather than an actual proven result. Let P' = conv(P $\bigcup$ Q') be the desired polytope construction.
Let Q+ be the set of all the faces (i.e., elements) of P having a non-empty intersection (i.e., at least a vertex) with Q. Let QC be the set of all the faces of P other than those of Q+; i.e., QC = FL(P) - Q+.
In terms of the f-vector, I believe you have f(P')(n) $\ge$ f(QC)(n) + 2f(Q+)(n) + f(Q+)(n-1) for n = 1 through d + 1 (the dimension of P'). The middle term corresponds to Q+ plus a copy of itself intersecting with Q' rather than with Q.
The last term represents n-dimensional elements having a facet within Q+, whose intersection with Q is duplicated in an intersection with Q'. For an (n-1)-face F of Q+ which is also an element of Q, this is a prism over F. For other (n-1)-faces F of Q+, the resulting n-face won't be a prism over F. In the case of the tridiagonal Birkhoff polytope, it appears that the above inequality is an equality, and there are two possible ways to generate the n-face from a given (n-1)-face (other than a prism.)
