LINEAR-APPROXIMATION IN L(N)(INFINITY)

The paper studies the approximation behavior of a linear subspace U in l(n)infinity; i.e., in R(n) equipped with the maximum norm. As a prin cipal tool the Plucker-Grassmann coordinates of U are used; they allow a classification of the index set {1, ..., n) through which we determ ine the extremal points of the intersection of the orthogonal compleme nt U(perpendicular-to) of U and the closed l(n)1-unit ball in R(n), le ading to the dual problem. As a consequence, we describe the metric co mplement U(0) of U and give a decomposition of R(n)\U into a finite se t of pairwise disjoint convex cones on which the metric projection P(U ) has some characteristic properties. In the Chebyshev case, e.g., the metric projection is linear on these cones and, consequently, globall y 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 P(U) with and without the Nulleigenschaft, and characterize those subspaces U w hich admit a linear selection. (C) 1994 Academic Press, Inc.