TITLE: Alexander's theorem (on stellar moves)
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: 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 $K'$.