On the dimension of the set of minimal projections

TL;DR

This paper investigates the dimension of the set of minimal projections in finite-dimensional normed spaces, providing bounds and properties, especially in polyhedral spaces, with implications for uniqueness and norming pairs.

Contribution

It establishes optimal bounds on the dimension of minimal projection sets and explores their properties in polyhedral normed spaces, including uniqueness and norming pairs.

Findings

Optimal upper bounds for the dimension of minimal projection sets.

In polyhedral spaces, minimal projections are generically unique.

Existence of minimal projections with many norming pairs in polyhedral spaces.

Abstract

Let $X$ be a finite-dimensional normed space and let $Y \subseteq X$ 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.