A Lower Bound for Shallow Partitions

Let P be a planar n-point set. A k-partition of P is a subdivision of P into n/k parts of roughly equal size and a sequence of triangles such that each part is contained in a triangle. A line is k-shallow if it has at most k points of P below it. The crossing number of a k-partition is the maximum number of triangles in the partition that any k-shallow line intersects. We give a lower bound of Omega(log (n/k)/loglog(n/k)) for this crossing number, answering a 20-year old question of Matousek.