CUTTINGS AND APPLICATIONS

Abstract

We prove a general lemma on the existence of (1/r)-cutting of geometric objects in E(d) that satisfy certain properties. We use this lemma to construct (1/r)-cuttings of small size for arrangements of line segments in the plane and arrangements of triangles in 3-space; for line segments in the plane we obtain a cutting of size O(r+Ar-2/n(2)), and for triangles in 3-space our cutting has size O(r(2+e)+Ar-3/n(3)). 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.