Metric projection and stratification of the Grassmannian (Q1977375)

This paper is based on a previous investigation of the author [J. Approximation Theory 76, No. 3, 326-350 (1994; Zbl 0802.41022)] of the classical problem of best approximation of \(l_\infty(n)\) (the Euclidean space \(\mathbb{R}^n\), \(n\in \mathbb{N}\), endowed with the maximum norm \(\|\cdot\|_\infty\)) by a linear subspace \(U\) using Plücker-Grassmann coordinates and a classification corresponding to the set of vertices of the polyhedron \(Q= U^\perp\cap \overline{b^l_1(0)}\). The author shows that: (a) the extremal points of \(Q\) are precisely the \(l_1(n)\)-normed elementary vectors in \(U^\perp\), which in turn correspond to all circuit vectors of a matrix the columns of which form a basis of the subspace, (b) the classification is discrete and divides the Grassmann manifold into finitely many strata using ideas from approximation theory, and (c) the stratification evidenced by (b) is identical with the decompositions of the Grassmannian given by \textit{I. M. Gel'fand}, \textit{R. M. Goresky}, \textit{R. D. MacPherson} and \textit{V. V. Serganova} [Adv. Math. 63, 301-316 (1987; Zbl 0622.57014)].