Almost minimal orthogonal projections

Abstract

The projection constant Π(E):= Π(E, ℓ∞) of a finite-dimensional Banach space E ⊂ ℓ∞ is by definition the smallest norm of a linear projection of ℓ∞ onto E. Fix n ≥ 1 and denote by Πn the maximal value of Π(·) amongst n-dimensional real Banach spaces. We prove for every ε > 0 that there exist an integer d ≥ 1 and an n-dimensional subspace E ⊂ d1 such that Πn ≤ Π(E,d1)+2ε and the orthogonal projection P : d1 → E is almost minimal in the sense that P ≤ Π(E,d 1) + ε. As a consequence of our main result, we obtain a formula relating Πn to smallest absolute value row-sums of orthogonal projection matrices of rank n. © 2021 Springer Nature Switzerland AG