4 added a weaker form of the question

It's well-known that any compact polyhedron $P$ in $\mathbb{R}^n$ (we talk about piecewise-linear setting there, i.e. $P$ is a finite union of compact convex polytopes) can be triangulated into (geometric) simplices, although sometimes it is necessary to add "extra points" in $P$ to serve as vertices of simplices in the triangulation $\mathcal{T}$. E.g. Schönhardt polyhedron requires such extra points. (Here By $\mathcal{T}$ we mean a partition of $P$ into finitely many simplices $T\in\mathcal{T}$ --- more precisely, the interiors $int(T)$ of $T$'s do not intersect, and the closure of $\cup_{T\in\mathcal{T}}int(T)$ equals $P$. The vertices of $T$'s that are not vertices of $P$ are these "extra points" we talk about.)

It looks correct that one can always construct such a $\mathcal{T}$ so that each extra point in it is "non-rigid", i.e. it can be continuously moved inside an open subset of the face of minimal dimension it is inserted into, so that after such a deformation $\mathcal{T}$ remains a triangulation of $P$. Is this indeed correct, and can anyone point out a reference?

Added: A weaker form of the question: show that each extra poing in $\mathcal{T}$ is not prescribed, i.e. for any vertex $y$ of $\mathcal{T}$ which is not a vertex of $P$ there exists another triangulation of $P$ which does not have $y$ as a vertex. [This still suffices for our purpose, of showing that a part of certain kind of moment generating functions, for moments of a uniform measure supported on $P$, does not depend upon $\mathcal{T}$ ]. This is easy to see that $y$ lying in the interior of $P$ is not prescribed---one can directly construct a new triangulation not using $y$, by choosing the points of intersection of the edges on $y$ with a sufficiently small sphere around $y$ and re-triangulating new convex pieces without using $y$. But it is not obvious for $y$ lying in a proper face of $P$.

3 clarified the openness condition

It's well-known that any compact polyhedron $P$ in $\mathbb{R}^n$ (we talk about piecewise-linear setting there, i.e. $P$ is a finite union of compact convex polytopes) can be triangulated into (geometric) simplices, although sometimes it is necessary to add "extra points" in $P$ to serve as vertices of simplices in the triangulation $\mathcal{T}$. E.g. Schönhardt polyhedron requires such extra points. (Here By $\mathcal{T}$ we mean a partition of $P$ into finitely many simplices $T\in\mathcal{T}$ --- more precisely, the interiors $int(T)$ of $T$'s do not intersect, and the closure of $\cup_{T\in\mathcal{T}}int(T)$ equals $P$. The vertices of $T$'s that are not vertices of $P$ are these "extra points" we talk about.)

It looks correct that one can always construct such a $\mathcal{T}$ so that each extra point in it is "non-rigid", i.e. it can be continuously moved inside an open set subset of the face of minimal dimension it is inserted into, so that after such a deformation $\mathcal{T}$ remains a triangulation of $P$. Is this indeed correct, and can anyone point out a reference?

2 clarified the situation considered (everything is piecewise-linear)

It's well-known that any compact polyhedron $P$ in $\mathbb{R}^n$ (we talk about piecewise-linear setting there, i.e. $P$ is a finite union of compact convex polytopes) can be triangulated into (geometric) simplices, although sometimes it is necessary to add "extra points" in $P$ to serve as vertices of simplices in the triangulation $\mathcal{T}$. E.g. Schönhardt polyhedron requires such extra points. (Here By $\mathcal{T}$ we mean a partition of $P$ into finitely many simplices $T\in\mathcal{T}$ --- more precisely, the interiors $int(T)$ of $T$'s do not intersect, and the closure of $\cup_{T\in\mathcal{T}}int(T)$ equals $P$. The vertices of $T$'s that are not vertices of $P$ are these "extra points" we talk about.)

It looks correct that one can always construct such a $\mathcal{T}$ so that each extra point in it is "non-rigid", i.e. it can be continuously moved inside an open set so that after such a deformation $\mathcal{T}$ remains a triangulation of $P$. Is this indeed correct, and can anyone point out a reference?

1