On the sum of squares of cell complexities in hyperplane arrangements

Boris Aronov, Jiří Matoušek, Micha Sharir

Research output: Contribution to journalArticlepeer-review

Abstract

Let H be a collection of n hyperplanes in Rd, d≥2. For each cell c of the arrangement of H let fi(c) denote the number of faces of c of dimension i, and let f(c) = ∑i=0d-1 fi(c). We prove that ∑c f(c)2 = O(ndlog d 2-1 n), where the sum extends over all cells of the arrangement. Among other applications, we show that the total number of faces bounding any m distinct cells in an arrangement of n hyperplanes in Rd is O(m 1 2n d 2log ( d 2-1) 2 n) and provide a lower bound on the maximum possible face count in m distinct cells, which is close to the upper bound, and for many values of m and n is Ω(m 1 2n d 2).

Original languageEnglish
Pages (from-to)311-321
Number of pages11
JournalJournal of Combinatorial Theory - Series A
Volume65
Issue number2
DOIs
StatePublished - Feb 1994

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