Convex Polygons Made from Few Lines and Convex Decompositions of Polyhedra

We give a worst-case bound of Θ(m2/3n2/3+ n) on the complexity of m convex polygons whose sides come from n lines. The same bound applies to the complexity of the horizon of a segment that intersects m faces in an incrementally-constructed erased arrangement of n lines. We also show that Chazelle's notch-cutting procedure, when applied to a polyhedron with n faces and r reflex dihedral angles, gives a convex decomposition with Θ(nr+r7/3) worst-case complexity.

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