### Abstract

Let H be a collection of n hyperplanes in R^{d}, d≥2. For each cell c of the arrangement of H let f_{i}(c) denote the number of faces of c of dimension i, and let f(c) = ∑_{i=0}
^{d-1} f_{i}(c). We prove that ∑_{c} f(c)^{2} = O(n^{d}log^{ 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 R^{d} is O(m^{ 1 2}n^{ d 2}log^{ ( 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 2}n^{ d 2}).

Original language | English (US) |
---|---|

Pages (from-to) | 311-321 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 65 |

Issue number | 2 |

DOIs | |

State | Published - 1994 |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*65*(2), 311-321. https://doi.org/10.1016/0097-3165(94)90027-2

**On the sum of squares of cell complexities in hyperplane arrangements.** / Aronov, Boris; Matoušek, Jiří; Sharir, Micha.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 65, no. 2, pp. 311-321. https://doi.org/10.1016/0097-3165(94)90027-2

}

TY - JOUR

T1 - On the sum of squares of cell complexities in hyperplane arrangements

AU - Aronov, Boris

AU - Matoušek, Jiří

AU - Sharir, Micha

PY - 1994

Y1 - 1994

N2 - 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=0 d-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).

AB - 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=0 d-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).

UR - http://www.scopus.com/inward/record.url?scp=38149147699&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149147699&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(94)90027-2

DO - 10.1016/0097-3165(94)90027-2

M3 - Article

AN - SCOPUS:38149147699

VL - 65

SP - 311

EP - 321

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -