论文信息 - On the zone of a surface in a hyperplane arrangement

On the zone of a surface in a hyperplane arrangement

LetH be a collection ofn hyperplanes in ℝd, letA denote the arrangement ofH, and let σ be a (d−1)-dimensional algebraic surface of low degree, or the boundary of a convex set in ℝd. Thezone of σ inA is the collection of cells ofA crossed by σ. We show that the total number of faces bounding the cells of the zone of σ isO(nd−1 logn). More generally, if σ has dimensionp, 0≤p<d, this quantity isO(n[(d+p)/2]) ford−p even andO(n[(d+p)/2] logn) ford−p odd. These bounds are tight within a logarithmic factor.

Micha Sharir | Boris Aronov | Marco Pellegrini

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