# On levels in arrangements of lines, segments, planes, and triangles

Published

Journal Article

We consider the problem of bounding the complexity of the k-th level in an arrangement of n curves or surfaces, a problem dual to, and extending, the well-known k-set problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n√k+1) for the complexity of the k-th level in an arrangement of n lines. (b) We derive an improved version of Lovasz Lemma in any dimension, and use it to prove a new bound, O(n2k2/3), on the complexity of the k-th level in an arrangement of n planes in R3, or on the number of k-sets in a set of n points in three dimensions. (c) We show that the complexity of any single level in an arrangement of n line segments in the plane is O(n3/2), and that the complexity of any single level in an arrangement of n triangles in 3-space is O(n17/6).

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Aronov, B; Sharir, M

### Published Date

- January 1, 1997

### Published In

- Proceedings of the Annual Symposium on Computational Geometry

### Start / End Page

- 30 - 38

### Digital Object Identifier (DOI)

- 10.1145/262839.262856

### Citation Source

- Scopus