Metric projection and stratification of the Grassmannian Original Research Article

M. Finzel [J. Approx. Theory 76 (1994) 326–350] investigated the classical problem of best approximation of l∞(n) by a linear subspace U using Plücker–Grassmann coordinates and a classification thereof. The classification corresponds to the set of vertices of the polyhedron image. Indeed, the extremal points of Q are precisely the l1(n)-normed elementary vectors in Uperpendicular, which in turn correspond to all circuit vectors of a matrix the columns of which form a basis of the subspace. The classification is discrete and divides the Grassmann manifold into finitely many strata using ideas from Approximation Theory. We identify this stratification with three decompositions of the Grassmannian identified by I.M.H. Gelfand et al. [Adv. Math. 63 (1987) 301–316].