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 $$E \subset \ell _1^d$$ such that $${\Pi _n} \le \Pi (E,\ell _1^d) + 2\varepsilon $$ and the orthogonal projection $$P:\ell _1^d \to E$$ is almost minimal in the sense that $$\left\| P \right\| \le \Pi (E,\ell _1^d) + \varepsilon $$ . 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.