Bounding the number of k-faces in arrangements of hyperplanes

Komei Fukuda; Shigemasa Saito; Akihisa Tamura; Takeshi Tokuyama

doi:10.1016/0166-218X(91)90067-7

Bounding the number of k-faces in arrangements of hyperplanes

Komei Fukuda, Shigemasa Saito, Akihisa Tamura, Takeshi Tokuyama

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

Abstract

We study certain structural problems of arrangements of hyperplanes in d-dimensional Euclidean space. Of special interest are nontrivial relations satisfied by the f-vector f=(f₀,f₁,...,f_d) of an arrangement, where f_k denotes the number of k-faces. The first result is that the mean number of (k-1)-faces lying on the boundary of a fixed k-face is less than 2k in any arrangement, which implies the simple linear inequality f_k>(d-k+1) kf_--1 if f_k≠0. Similar results hold for spherical arrangements and oriented matroids. We also show that the f-vector and the h-vector of a simple arrangement is logarithmic concave, and hence unimodal.

Original language	English
Pages (from-to)	151-165
Number of pages	15
Journal	Discrete Applied Mathematics
Volume	31
Issue number	2
DOIs	https://doi.org/10.1016/0166-218X(91)90067-7
Publication status	Published - 1991 Apr 15
Externally published	Yes

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Applied Mathematics

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abstract = "We study certain structural problems of arrangements of hyperplanes in d-dimensional Euclidean space. Of special interest are nontrivial relations satisfied by the f-vector f=(f0,f1,...,fd) of an arrangement, where fk denotes the number of k-faces. The first result is that the mean number of (k-1)-faces lying on the boundary of a fixed k-face is less than 2k in any arrangement, which implies the simple linear inequality fk>(d-k+1) kf--1 if fk≠0. Similar results hold for spherical arrangements and oriented matroids. We also show that the f-vector and the h-vector of a simple arrangement is logarithmic concave, and hence unimodal.",

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T1 - Bounding the number of k-faces in arrangements of hyperplanes

AU - Fukuda, Komei

AU - Saito, Shigemasa

AU - Tamura, Akihisa

AU - Tokuyama, Takeshi

PY - 1991/4/15

Y1 - 1991/4/15

N2 - We study certain structural problems of arrangements of hyperplanes in d-dimensional Euclidean space. Of special interest are nontrivial relations satisfied by the f-vector f=(f0,f1,...,fd) of an arrangement, where fk denotes the number of k-faces. The first result is that the mean number of (k-1)-faces lying on the boundary of a fixed k-face is less than 2k in any arrangement, which implies the simple linear inequality fk>(d-k+1) kf--1 if fk≠0. Similar results hold for spherical arrangements and oriented matroids. We also show that the f-vector and the h-vector of a simple arrangement is logarithmic concave, and hence unimodal.

AB - We study certain structural problems of arrangements of hyperplanes in d-dimensional Euclidean space. Of special interest are nontrivial relations satisfied by the f-vector f=(f0,f1,...,fd) of an arrangement, where fk denotes the number of k-faces. The first result is that the mean number of (k-1)-faces lying on the boundary of a fixed k-face is less than 2k in any arrangement, which implies the simple linear inequality fk>(d-k+1) kf--1 if fk≠0. Similar results hold for spherical arrangements and oriented matroids. We also show that the f-vector and the h-vector of a simple arrangement is logarithmic concave, and hence unimodal.

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Bounding the number of k-faces in arrangements of hyperplanes

Abstract

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