, Volume 7, Issue 1, pp 359-369

Counting facets and incidences

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We show thatm distinct cells in an arrangement ofn planes in ℝ3 are bounded byO(m 2/3 n+n 2) faces, which in turn yields a tight bound on the maximum number of facets boundingm cells in an arrangement ofn hyperplanes in ℝ d , for everyd≥3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in ℝ3. We also present a simpler proof of theO(m 2/3 n d/3+n d−1) bound on the number of incidences betweenn hyperplanes in ℝ d andm vertices of their arrangement.

Work on this paper was supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center Grant NSF-STC88-09648, and by NSA Grant MDA 904-89-H-2030.