Acute triangulations of polyhedra and ℝN
We study the problem of acute triangulations of convex polyhedra and the space ℝn. 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 ℝn do not exist for n≥5. In the opposite direction, in ℝ3, 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 ℝ4 if all dihedral angles are bounded away from π/2.
Mathematics Subject Classification (2000)52B05 52C17 51M20 52B10 52C22
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