The depth of Tsirelson's norm

Abstract

Tsirelson's norm $\Vert \cdot {\Vert }_T$ on $c_{00}$ is defined as the supremum over a certain collection of iteratively defined, monotone increasing norms $\Vert \cdot {\Vert }_k$. 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 $c_{00}$), $\Vert x{\Vert }_T=\Vert x{\Vert }_k$. 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.

26 pages, 1 figure. Comments welcome