Cuttings and applications

We prove a general lemma on the existence of {I/r)-cuttings of geometric objects in Ed that satisfy certain properties. We use this lemma to construct (I/r)-cuttings of (asymptotically) optimal size for arrangements of line segments in the plane and arrangements oftriangles in 3-space; for line segments in the plane we obtain a cutting of size O(r + Ar2/n2 ), and for triangles in 3-space our cutting has size O(r2a(r) + Ar /n3 ). Here A is the combinatorial complexity of the arrangement. Finally, we use these results to obtain new results for several problems concerning line segments in the plane and triangles in 3-space.