A note on hyper-plane arrangements in Rd

Nhattrieu Duong; Moshe Idan; Rom Pinchasi; Jason L. Speyer

doi:10.47443/dml.2021.0048

Author(s)

Nhattrieu Duong Department of Mechanical and Aerospace Engineering, University of California,Los Angeles,USA

Moshe Idan Faculty of Aerospace Engineering, Technion - Israel Institute of Technology,Haifa,Israel

Rom Pinchasi Faculty of Mathematics, Technion - Israel Institute of Technology,Haifa,Israel

Jason L. Speyer Department of Mechanical and Aerospace Engineering, University of California,Los Angeles,USA

Abstract

This note addresses hyper-plane arrangements in Rd and functions that are constant in the interior of each of the d-dimensional faces of the arrangement. We show that such a function g can be expressed in a simple form using basis functions that are products of d or less indicator functions of the open half-spaces bounded by the hyper-planes in the arrangement. Moreover, we present a simple and efficient algorithm that can be used to express g as a linear combination of these basis functions.

Publication Details

Page(s) 79-85

DOI 10.47443/dml.2021.0048

Published Journal: Discrete Mathematics Letters, Volume: 7, Issue: 0, Year: 2020

Request the full text from PASTIC

Keywords

hyperplanes, hyperplane-arrangements, linear basis

References

Fern J.,Speyer J. L.,Idan M. .2015 .Stochastic estimation for two-state linear dynamic systems with additive Cauchy noises. IEEE Trans. Automat. Control, 60 : 3367-3372.

Ferna J.,Speyer J. L.,Idan M. .2015 .Stochastic control for linear systems with additive Cauchy noises. IEEE Trans. Automat. Control, 60 : 3373-3378.

Goodman J. E.,J. O'Rourke J. E. .2018 .. Handbook of Discrete and Computational Geometry, Third Edition, : .

Idan M.,Speyer J. L. .2014 .Multivariate Cauchy estimator with scalar measurent and process noises. SIAM J. Control Optim, 52 : 1108-1141.

Stanley R.,2nd Edition,Hyperplane Arrangements .2011 .. , 3 : .

Zaslavsky T. .1975 .Face-Count Formulas for Partitions of Space by Hyperplanes, Mem. Amer. Math. Soc., : .