Median and center hyperplanes in Minkowski spaces--a unified approach

In this paper we will extend two known location problems from Euclidean n-space to all n-dimensional normed spaces, n¿ 2. Let X be a "nite set of weighted points whose a2ne hull is n-dimensional. Our "rst objective is to "nd a hyperplane minimizing (among all hyperplanes) the sum of weighted distances with respect to X. Such a hyperplane is called a median hyperplane with respect to X, and we will show that for all distance measures d derived from norms one of the median hyperplanes is the a2ne hull of n of the demand points. (This approach was already presented in the recent survey (Discrete Appl. Math. 89 (1998) 181), but without proofs. Here we give the complete proofs to all necessary lemmas.) On the other hand, we will prove that one of the hyperplanes minimizing (among all hyperplanes) the maximum weighted distance to some point from X has the same maximal distance to least n+1 a2nely independent demand points (such a hyperplane is said to be a center hyperplane of X). Both these results allow polynomially bounded algorithmical approaches to median and center hyperplanes and the respective distance sums or maximal distances for any "xed dimension n¿ 2, and in particular we discuss the algorithms for both the problems in the case of polyhedral norms. Also two independence of norm results for optimal hyperplanes with "xed slope will be derived, and "nally the considerations are even extended to gauges which are no longer combined with a norm. c © 2001 Elsevier Science B.V. All rights reserved.