N-dimensional Normed Space Research Articles

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 find a hyperplane H minimizing the weighted sum f( H) of distances to a given, finite 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 affine hull of n of the demand points and, moreover, that each median hyperplane is a halving one (in a sense defined below) with respect to the given point set. Also an independence of norm result for finding optimal hyperplanes with fixed 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 finally a characterization of all smooth norms by a sharpened incidence criterion for median hyperplanes is mentioned.

Operators with large trace, and a characterization of

Let E be a real or complex n-dimensional normed space. Deschaseaux (1) has shown that if the absolutely summing constant π1(E) = n then E is isometrically isomorphic to . (For the definition of π1(E) and related quantities, see (2).) It is well-known ((3), (4) and (6)) that if E is a P1-space then E is isometrically isomorphic to . We shall show that Deschaseaux' result follows easily from this, and from the following elementary result: Proposition. If S is a linear operator on a real or complex n-dimensional normed space E, then |trace S| ≤ ‖S‖, with equality if and only if S is a scalar multiple of the identity I.