How can pushing a vertex in a polytope lead to merging facets?

Question

I'm trying to understand the Lemma 2.2 and Corollary 2.3 of Francisco Santos paper "A counterexample to the Hirsch conjecture": http://arxiv.org/abs/1006.2814

Corollary 2.3 is a proof of a result of Klee saying: "For every polytope Q there is a simplicial polytope Q of the same dimension and number of vertices and with the same or greater dual diameter."

The definiton of pushing is the following: We say that a polytope $Q'$ is obtained from $Q$ by pushing $v$ if the vertices of $Q'$ are $ V \setminus \lbrace v \rbrace \cup \lbrace v' \rbrace $ for a certain point $v' \in Q$ and the only hyperplanes spanned by vertices of $Q$ that intersect the segment $vv'$ are those containing $v$.

For proving that corollary the following Lemma 2.2 is stated and proofed:

Let $Q'$ be obtained from $Q$ by pushing $v$. Then:

1. Let $F'$ be a facet of $Q$ with vertex set $S$ and let $S = S' \setminus \lbrace v' \rbrace \cup \lbrace v \rbrace$ or $S = S'$ depending on whether $v' \in F'$ or not. Then, there is a unique facet $\phi(F')$ of $Q$ such that $S \subset \phi(F' )$.
2. The map $F' → \phi(F')$ sends adjacent facets of $Q'$ to either the same or adjacent facets of $Q$. (That is, $\phi$ is a simplicial map between the dual graphs of $Q'$ and $Q$).

In the definition of the Lemma and also in its proof (see the paper, especially te sentence proofing part 2) it is mentioned that at the "beginning" of the pushing it can happen that two facets merge. I wasn't able to find an example how this can happen. Can anybody explain to me how this is possible?

I also don't get how The Corollary (which is actually the interesting part) is proofed from that Lemma, but i think this has to do with not understanding how this merging of facets happens.

Answer 1

What happens as you push $v$, is that you can break a face into two. Start with a cube in $\mathbb R^3$, choose a vertex, and start to push it into the interior of the cube. Each of the three facets of the cube containing $v$ will be broken into two triangles.

The "merging" happens if you think about reversing this procedure (moving $v'$ back to $v$). Santos says that he is thinking about it this way at the beginning of the proof.