On the dimension of the set of minimal projections

Abstract

Let X be a finite-dimensional normed space and let Y⊆X be its proper linear subspace. The set of all minimal projections from X to Y is a convex subset of the space all linear operators from X to X and we can consider its affine dimension. We establish several results on the possible values of this dimension. We prove optimal upper bounds in terms of the dimensions of X and Y. Moreover, we improve these estimates in the polyhedral normed spaces for an open and dense subset of subspaces of the given dimension. As a consequence, in the polyhedral normed spaces a minimal projection is unique for an open and dense subset of hyperplanes. To prove this, we establish certain new properties of the Chalmers-Metcalf operator. Another consequence is the fact, that for every subspace of a polyhedral normed space, there exists a minimal projection with many norming pairs.