On Constants for Cuttings in the Plane

Abstract. A theorem of Chazelle and Friedman with numerous applications in combinatorial and computational geometry asserts that for any set L of n lines in the plane and for any parameter r>1 there exists a subdivision of the plane into at most Cr2 (possibly unbounded) triangles, C a constant, such that the interior of each triangle is intersected by at most n/r lines of L . (Such a subdivision is called a (1/r) -cutting for L .) We give upper and lower bounds on the constant C . We also consider the canonical triangulation of the arrangement of a random sample of r lines from L . Although this typically is not a (1/r) -cutting, the expectation of the k th degree average of the number of lines intersecting a triangle is O(n/r) for any fixed k . We estimate the constant of proportionality in this result.

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