Acute Triangulations of Polyhedra and R

We study the problem of acute triangulations of convex polyhedra and the space R. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n≥4. Further, we prove that acute triangulations of the space R do not exist for n≥5. In the opposite direction, in R, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of R if all dihedral angles are bounded away from π/2.