If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?
In particular, is this true in $R^{4}$? In 2 and 3 dimensions any two polyhedra
that are homeomorphic are PL-homeomorphic, by theorems of Rado and Moise. In dimension $\geq 5$, this is a trivial special case of theorem 1.1 in 
M.A. Armstrong "The Hauptvermutung According to Lashof and Rothenberg" in 
*The Hauptvermutung Book*. But I have not found a statement that covers it for dimension 4; and I am not confident that dimension 4 can easily be reduced to
dimension 5. 


Also, if anyone can suggest a reference for this particular case that does not go through these very high-powered, difficult, general theorems, I would be interested on stylistic grounds. 



If a polyhedron is homeomorphic to a simplex, is it piecewise-linear homeomorphic?
If you assume that your polyhedron has only finite number of faces, I think the answer to your question is unknown. Moreover any answer to such question would give a solution to Smooth Poincare conjecture in dimension 4, which is still open. 


Indeed, suppose you have a four-dimensional sphere with an exotic smooth structure. Then you can always triangulate such a sphere in a finite number of simplexes. Now, throw away a simplex from such a triangulation. What you get is a homeomorphic to a simplex, but can not be PL diffeomorphic to it, otherwise your initial sphere would be PL diffeomerphic to the standard one, which is not sow since you sphere is exotic. 


