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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:

  1. If Q is a vertex, then conv(P $\bigcup$ Q') is a pyramid over P.

  2. If Q = P, then conv(P $\bigcup$ Q') is a prism over P (i.e., a rectangular product of P and a line segment).

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Motivation Part 1: For brevity, denote the vertices of the tridiagonal Birkhoff polytope $\Omega^t_n$ using cycle notation rather than matrix representation. Note that the facet of $\Omega^t_n$ at $a_{n-1,n}$ =$a_{n,n-1}$ = 0 is $\Omega^t_{n-1}$, with vertices at all compositions of disjoint cycles (12), (23)… (n-2 n-1) (including the identity permutation). – Dan Moore Sep 27 '11 at 14:37
Motivation continued: Consider the translation T: v → v(n n+1); v Є V, the set of vertices of $\Omega^t_{n-1}$. The set of translated vertices VT subtends a polytope outside the affine hull of $\Omega^t_n$ of combinatorial type $\Omega^t_{n-1}$. Also, conv($\Omega^t_n$, VT) = $\Omega^t_{n+1}$ – Dan Moore Sep 27 '11 at 14:40

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.)

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