Linear-Approximation in l∞n Original Research Article

The paper studies the approximation behavior of a linear subspace U in l∞n; i.e., in Rn equipped with the maximum norm. As a principal tool the Plücker-Graßmann coordinates of U are used; they allow a classification of the index set {1, ..., n} through which we determine the extremal points of the intersection of the orthogonal complement U⊥ of U and the closed l1n-unit ball in Rn, leading to the dual problem. As a consequence, we describe the metric complement U(0) of U and give a decomposition of Rn\Uinto a finite set of pairwise disjoint convex cones on which the metric projection PU has some characteristic properties. In the Chebyshev case, e.g., the metric projection is linear on these cones and, consequently, globally lipschitz continuous. A refinement allows an analogous statement for the strict approximation, proving a conjecture of Wu Li. Besides the strict approximation, were studying continuous selections of PU with and without the Nulleigenschaft, and characterize those subspaces U which admit a linear selection.