Invariants in convex geometry and Banach space theory

2. Cotype of Projective Tensor Spaces

A very easy fact about cotype and projective tensor spaces is that $L_1\otimes_{\pi} X$ always inherits the cotype of $X$. However, in general these questions ask for highly non-trivial methods and quite involved tools.

In 1974, Tomczak-Jaegermann proved that the projective tensor space $\ell_2\otimes_{\pi}\ell_2$ has cotype 2. More than 15 years later, Pisier showed that if $p,q\in[2,\infty)$, then $L_p\otimes_{\pi} L_q$ has cotype $\max\{p,q\}$.

In a recent paper, Briet, Naor and Regev showed that the projective triple tensor spaces $\ell_p\otimes_{\pi}\ell_q\otimes_{\pi}\ell_r$, $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} \leq 1$ fail to have non-trivial cotype, i.e., they contain $\ell_{\infty}^n$’s uniformly. The main tool they use are locally decodable codes of subexponetial length and it comes from theoretical computer science. The initial idea was to use tools from Banach space theory to obtain new estimates on the length of these codes. The authors were also able to prove that for every $p\in(1,\infty)$ there exists a Banach space $X$ of finite cotype such that $\ell_p\otimes_{\pi} X$ fails to have finite cotype.