Stability of maximal relative projection constants

Abstract

For positive integers $n\ge r$, let $\lambda (r,n)$ denote the \emph{maximal relative projection constant} of $r$-dimensional subspaces of ${\ell }_{\infty }^n$ and $\lambda (r)$ denote the \emph{maximal absolute projection constant}, respectively. It is known that for any fixed $r$, $\lambda (r,n)$ is a non-decreasing sequence with limit $\lambda (r)$ as $n\to \infty$. A natural question is whether $\lambda (r,n)$ stabilizes at $\lambda (r)$ for some $n>r$. We prove that for any fixed $r$, \[λ(r,n)=λ(r) \qquad \text{for every}\qquad n\ge 2^{r}\binom{r+1}{2}.\] This answers a question of Basso. The technique used is of independent interest.