Almost minimal orthogonal projections

Abstract

The projection constant $\Pi (E):=\Pi (E,{\ell }_{\infty })$ of a finite-dimensional Banach space $E\subset {\ell }_{\infty }$ is by definition the smallest norm of a linear projection of ${\ell }_{\infty }$ onto $E$. Fix $n\ge 1$ and denote by ${\Pi }_n$ the maximal value of $\Pi (\cdot )$ amongst $n$-dimensional real Banach spaces. We prove for every $\epsilon >0$ that there exist an integer $d\ge 1$ and an $n$-dimensional subspace $E\subset {\ell }_1^d$ such that ${\Pi }_n\le \Pi (E,{\ell }_1^d)+2\epsilon$ and the orthogonal projection $P:{\ell }_1^d\to E$ is almost minimal in the sense that $\Vert P\Vert \le \Pi (E,{\ell }_1^d)+\epsilon$. As a consequence of our main result, we obtain a formula relating ${\Pi }_n$ to smallest absolute value row-sums of orthogonal projection matrices of rank $n$.

final version

References in corpus (1)

• Computation of maximal projection constants

Cited by in corpus (2)

• On the value of the fifth maximal projection constant
• Erratum to "Computation of maximal projection constants"