Partitioning axis-parallel lines in 3D
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Let L be a set of n axis-parallel lines in ℝ^3. We are are interested in partitions of ℝ^3 by a set H of three planes such that each open cell in the arrangement 𝒜(H) is intersected by as few lines from L as possible. We study such partitions in three settings, depending on the type of splitting planes that we allow. We obtain the following results. ∙ There are sets L of n axis-parallel lines such that, for any set H of three splitting planes, there is an open cell in 𝒜(H) that intersects at least ⌊ n/3 ⌋-1 ≈1/3n lines. ∙ If we require the splitting planes to be axis-parallel, then there are sets L of n axis-parallel lines such that, for any set H of three splitting planes, there is an open cell in 𝒜(H) that intersects at least 3/2⌊ n/4 ⌋ -1 ≈( 1/3+1/24) n lines. Furthermore, for any set L of n axis-parallel lines, there exists a set H of three axis-parallel splitting planes such that each open cell in 𝒜(H) intersects at most 7/18 n = ( 1/3+1/18) n lines. ∙ For any set L of n axis-parallel lines, there exists a set H of three axis-parallel and mutually orthogonal splitting planes, such that each open cell in 𝒜(H) intersects at most ⌈5/12 n ⌉≈( 1/3+1/12) n lines.
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