The depth of Tsirelson’s norm

Tsirelson’s norm $\|\cdot \|_T$ on $c_{00}$ is defined as the supremum over a certain collection of iteratively defined, 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 modifications of Tsirelson’s original norm.