I am wondering which relation(s) there are between triangulations of combinatorially equivalent polytopes that use no new points:
Let $P,Q$ be a $n$-polytopes such that their face lattices are isomorphic (say via $\phi$), that is $P,Q$ are combinatorially equivalent. Suppose we have an ordering on the vertices of $P,Q$ such that the vertices are labeled in such a way that an element $\{p_{i_1},\ldots,p_{i_k}\}$ of the face lattice of $P$ corresponds to the element $\{q_{i_1},\ldots,q_{i_k}\}$ in the the face lattice of $Q$ under the combinatorial isomorphism $\phi$.
Let $T$ be a triangulation of $P$ that uses just the vertices of $P$ as vertices of its elements (say given by the sequence of vertices of simplices in $T$). Is $\phi(T)$ (given by the sequence of images of vertices of simplices in $T$) a triangulation of $Q$? This should imply that combinatorially equivalent polytopes have the "same" triangulations (that are using only vertices).
My main concern would be: How would I be sure that the so defined simplices in $Q$ do intersect properly?
This question kinda gets me, because I fail to find an explicit answer on the web and in the literature, despite seeming to be a rather straight forward question. I have (among others) looked into the books of Grünbaum, Ziegler and the "Triangulations" book by De Loera et al., as well as certain research articles. I am not entirely sure if it is some sort of obvious thing that I just dont see, or if there is actually some difficulty around.
What I looked into in some detail is:
In the "Triangulations" book I did find the following (Corollary 4.1.44): Two point configurations that have the same oriented matroids do have the same set of triangulations (actually the same set of polyhedral subdivisions), using only points in the given configuration.
Now the latter kind of "combinatorial equivalence" is finer then the earlier one (i.e. Example 6.3. in Zieglers book). I really wonder whether just the combinatorial data of the face lattices would suffice, or if there are counter examples? I just feel like the face lattice should not be enough, that there should be some "metric" necessitites around.
I also get that if two polytopes as above "share" a triangulation as above, there will be a piecewise linear homeomorphism linking both, that should get us combinatorial equivalence. To which extent can one reverse that?
Of course I came across the notion of secondary (and universal) polytopes, but at the moment i am still confused about what to read of them (and if feel like that this is a bit to heavy machinery to be necessary to get an answer to my question?).