Median Hyperplanes in Normed Spaces - A Survey

In this survey we deal with the location of hyperplanes in n{ dimensional normed spaces, i.e., we present all known results and a unifying approach to the so-called median hyperplane problem in Minkowski spaces. We describe how to nd a hyperplane H minimizing the weighted sum f(H) of distances to a given, nite set of demand points. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane. After summarizing the known results for the Euclidean and rectangular situation , we show that for all distance measures d derived from norms one of the hyperplanes minimizing f(H) is the aane hull of n of the demand points and, moreover, that each median hyperplane is a halving one (in a sense deened below) with respect to the given point set. Also an independence of norm result for nd-ing optimal hyperplanes with xed slope will be given. Furthermore we discuss how these geometric criteria can be used for algorithmical approaches to median hyperplanes, with an extra discussion for the case of polyhedral norms. And nally a characterization of all smooth norms by a sharpened incidence criterion for median hyperplanes is mentioned.