Alexander's theorem (on stellar moves) A widely used theorem these days says that given (abstract) simplicial complexes $K$ and $K'$, a polyhedron $P\subset R^n$ and homeomorphisms $|K|\to P$ and $|K'| \to P$ which are linear (better to say affine) in each simplex then $K$ and $K'$ are stellar equivalent as abstract simplicial complexes.
Therefore we have a sequence:
$$K \to K_1 \to K_2 \to K_3 \to \dots \to K_n\to K' $$
of abstract simplicial complexes where $K_{i+1}$ is obtained from $K_i$ by an isomorphism ("vertex name change"), a stellar subdivision or a welding (inverse stellar subdivision).
Question: can you do this in order that each $K_i$ is linearly (as opposed to piecewise linearly) embedded in $P$?
My version of the answer. Looking at the classical references of the proof of Alexander's theorems it does appear that the proofs do not imply this stronger statement: the problem being that sometimes $K_i$ may very well contain the star of a non-convex cell (which my be not "star-shaped") in $P$ and some welds may be not-realizable in $P$ without modifying $P$, slightly, to an isomorphic complex: c.f. L. C. Glaser, Geometrical combinatorial topology volume 1, pages 23 and 29.
PS: Great texts on Alexander's theorem are:
a) The first two chapters of  L. C. Glaser, Geometrical combinatorial topology volume 1.
b) Alexander's original paper: Alexander, James W. The combinatorial theory of complexes. Ann. of Math. (2) 31 (1930), no. 2, 292–320. 
c) Lickorish, W. B. R.: Simplicial moves on complexes and manifolds. (English summary) Proceedings of the Kirbyfest (Berkeley, CA, 1998), 299–320 (electronic), Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999. 
 A: Correct question. May be this quick recollections are completely false. I was thinking many years ago about it. Seems that this is a matter of arithmetic (!) of vertices of triangulations. It is visionabe when we are understanding triangulations as related to rational (!) fans of toric varieties and actually in rational case it is equivalent to "weak factorization conjecture" of equivariant birational isomormism of non-singular toric varietis in a sequence of blowups and blowdowns. This conjecture is supposed to be true.  So generally one should infinitesimally move triangulations  to rational, but this is far from innocent operation. Yo may need to heavily break P and actually a lot of interesting troubles come.
The story was developed around the failure of R. Morelli's proof of Oda-Hironaka strong factorisation conjecture, see for last list references here
https://arxiv.org/abs/0911.4693 
These days we have a lot of developments around algebraic geometry, physics and combinatorics of fans related to "Okunkov bodies". So perhaps one can use it for the question in an interesting way. 
A: Yes, two triangulations of a polyhedron can be connected by a sequence of geometric stellar moves (subdivisions and weldings). And you are right, this does not follow from the Alexander theorem, which deals with combinatorial stellar moves. This was proved by [Morelli, The birational geometry of toric varieties J. Algebraic Geom. 5(1996), no. 4, 751–782.] and [Wlodarczyk, Decomposition of birational toric maps in blow-ups & blow-downs, Trans. Amer. Math. Soc. 349 (1997), no. 1, 373–411.] (They proved also more: if the triangulation is unimodular, then there is a path through unimodular triangulation.)
An outline of Morelli's proof can be found in our article with Jean-Marc Schlenker: On the infinitesimal rigidity of polyhedra with vertices in convex position, here is the arXiv version.
Morelli-Wlodarczyk theorem deals with convex polyhedra.
It is not completely clear to me how to extend the theorem to non-convex polyhedra. If we modify the triangulations of the cells in some order, there can appear problems on the boundaries between the cells.
Maybe one can do the following: extend some stellar subdivisions of $K$ and $K'$ to triangulations $\overline{K}$ and $\overline{K'}$ of the convex hull of $P$. A sequence of stellar moves from $\overline{K}$ to $\overline{K'}$ induces a sequence of stellar moves between the stellar subdivisions of $K$ and $K'$. Hence we also have a sequence of moves joining $K$ and $K'$.
