The depth of Tsirelson's norm

Abstract

Tsirelson's norm $\|\cdot \|_T$ on $c_{00}$ is defined as the supremum over a certain collection of iteratively defined, monotone increasing norms $\|\cdot \|_k$. For each positive integer $n$, the value $j(n)$ is the least integer $k$ such that for all $x \in \mathbb{R}^n$ (here $\mathbb{R}^n$ is considered as a subspace of $c_{00}$), $\|x\|_T = \|x\|_k$. In 1989 Casazza and Shura asked what is the order of magnitude of $j(n)$. It is known that $j(n) \in \mathcal{O}(\sqrt{n})$. We show that this bound is tight, that is, $j(n) \in \Omega(\sqrt{n})$. Moreover, we compute the tight order of magnitude for some norms being modifications of the original Tsirelson's norm.