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Median hyperplanes in normed spaces

  • In this paper we deal with the location of hyperplanes in n-dimensional normed spaces. If d is a distance measure, our objective is to find a hyperplane H which minimizes f(H) = sum_{m=1}^{M} w_{m}d(x_m,H), where w_m ge 0 are non-negative weights, x_m in R^n, m=1, ... ,M demand points and d(x_m,H)=min_{z in H} d(x_m,z) is the distance from x_m to the hyperplane H. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane. We show that for all distance measures d derived from norms, one of the hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median hyperplane is (ina certain sense) a halving one with respect to the given point set.

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Author:Anita Schöbel, H. Martini
URN (permanent link):urn:nbn:de:hbz:386-kluedo-4590
Serie (Series number):Report in Wirtschaftsmathematik (WIMA Report) (18)
Document Type:Preprint
Language of publication:English
Year of Completion:1999
Year of Publication:1999
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2000/04/03
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011