Erased arrangements of lines and convex decompositions of polyhedra

In 1984, B. Chazelle [SIAM J. Comp., 13 (1984), pp. 488{507] proposed a notch-cutting procedure for decomposing a non-convex polyhedron into convex pieces. This paper shows that notch-cutting, when applied to a polyhedron with n faces and r re ex dihedral angles, gives a convex decomposition with (nr + r7=3) worst-case complexity. The upper and lower bounds are obtained by studying the complexity of the horizon of a segment in an incrementally-constructed erased arrangement of n lines. E cient deterministic algorithms to compute this decomposition are also described.