TY - JOUR

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

AU - Fukuda, Komei

AU - Saito, Shigemasa

AU - Tamura, Akihisa

AU - Tokuyama, Takeshi

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

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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U2 - 10.1016/0166-218X(91)90067-7

DO - 10.1016/0166-218X(91)90067-7

M3 - Article

AN - SCOPUS:0004515166

VL - 31

SP - 151

EP - 165

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 2

ER -