Stability of maximal relative projection constants

Abstract: For positive integers $n\ge r$, let $λ(r,n)$ denote the \emph{maximal relative projection constant} of $r$-dimensional subspaces of $\ell_\infty<sup>n$ and $λ(r)$ denote the \emph{maximal absolute projection constant}, respectively. It is known that for any fixed $r$, $λ(r,n)$ is a non-decreasing sequence with limit $λ(r)$ as $n\to \infty$. A natural question is whether $λ(r,n)$ stabilizes at $λ(r)$ for some $n&gt;r$. 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.