I'm thinking about such question.
It there a convex polyhedron $P$ that is not a union of disjoint simplices which vertices are vertices of $P$?
It's not true for convex polygons. I belive that answer is "yes", but can't imagine a proper example.
P.S. The question arose when some of my students tried to prove Carathéodory's theorem in $\mathbb{R}^3$.