Depth in an Arrangement of Hyperplanes View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

1999-09

AUTHORS

P. J. Rousseeuw, M. Hubert

ABSTRACT

A collection of n hyperplanes in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\Bbb R}$ \end{document}d forms a hyperplane arrangement. The depth of a point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\theta \in {\Bbb R}^d$ \end{document} is the smallest number of hyperplanes crossed by any ray emanating from θ . For d=2 we prove that there always exists a point θ with depth at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lceil n/3\rceil$ \end{document} . For higher dimensions we conjecture that the maximal depth is at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lceil n/(d+1)\rceil$ \end{document} . For arrangements in general position, an upper bound on the maximal depth is also established. Finally, we discuss algorithms to compute points with maximal depth. More... »

PAGES

167-176

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/pl00009452

DOI

http://dx.doi.org/10.1007/pl00009452

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1038466844


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