# Cotype of the triple projective tensor product of Hilbert spaces

The projective tensor product of two copies of $\ell_2$ is the space of trace-class operators, which has cotype 2, a strong regularity property of Banach spaces. For three factors, other combinations of $\ell_p$ spaces are known to lose all non-trivial cotype, a fact proved through locally decodable codes from computer science. Whether $\ell_2 \otimes_\pi \ell_2 \otimes_\pi \ell_2$ keeps some finite cotype is a long-standing question of Pisier.

## Problem

A Banach space $X$ has (Rademacher) *cotype* $q \in [2, \infty)$ if there is a constant $C$ such that for all finite sequences $x_1, \dots, x_m \in X$,
$$\Big(\sum_{i=1}^m \|x_i\|^q\Big)^{1/q} \le C\, \Big(\mathbb{E}\,\Big\|\sum_{i=1}^m \varepsilon_i x_i\Big\|^2\Big)^{1/2},$$
where $\varepsilon_1, \dots, \varepsilon_m$ are independent random signs. $X$ has *non-trivial cotype* if it has cotype $q$ for some $q < \infty$; by the Maurey–Pisier theorem this fails exactly when $X$ contains the spaces $\ell_\infty^n$ uniformly. The projective tensor norm on $X \otimes Y$ is
$$\|u\|_\pi = \inf\Big\{\sum_k \|x_k\|\,\|y_k\| : u = \sum_k x_k \otimes y_k\Big\},$$
and $X \otimes_\pi Y$ is the completion.

**Question (Pisier).** Does $\ell_2 \otimes_\pi \ell_2 \otimes_\pi \ell_2$ have non-trivial cotype?

## Background

The problem was recorded by Gilles Pisier at the AIM workshop on invariants in convex geometry and Banach space theory. Tensor products are a basic source of examples in Banach space theory, and cotype governs, for instance, how random sums behave in a space. As the source recounts, the link with coding theory arose from an attempt to use Banach space methods to bound the length of locally decodable codes.

## What is known

- Tomczak-Jaegermann (1974): $\ell_2 \otimes_\pi \ell_2$, the trace class, has cotype 2. Pisier later showed that $L_p \otimes_\pi L_q$ has cotype $\max\{p, q\}$ for $p, q \in [2, \infty)$ (both as recorded in the source).
- Briët, Naor and Regev (2012): if $p_1, p_2, p_3 \in (1, \infty)$ satisfy $1/p_1 + 1/p_2 + 1/p_3 \le 1$, then $\ell_{p_1} \otimes_\pi \ell_{p_2} \otimes_\pi \ell_{p_3}$ fails to have finite cotype; they also show that for every $p \in (1, \infty)$ there is a space $X$ of finite cotype such that $\ell_p \otimes_\pi X$ does not. The proof goes through locally decodable codes. The case $p_1 = p_2 = p_3 = 2$, where the sum of reciprocals is $3/2$, is not covered.
- Giladi, Prochno, Schütt, Tomczak-Jaegermann and Werner (2017) computed volume ratios of $\ell_p^n \otimes_\pi \ell_q^n \otimes_\pi \ell_r^n$ and derived information on the cotype 2 constants of these spaces.

## What would count as progress

- A proof that $\ell_2 \otimes_\pi \ell_2 \otimes_\pi \ell_2$ has cotype $q$ for some finite $q$, or that it contains $\ell_\infty^n$ uniformly (for instance via new lower bounds for locally decodable codes in the relevant regime).
- Quantitative estimates: the growth in $n$ of the cotype constants of $\ell_2^n \otimes_\pi \ell_2^n \otimes_\pi \ell_2^n$. Lower bounds can be supported by explicit families of tensors, whose norms can be estimated numerically for small $n$.
- The companion question from the same workshop: does $L_p \otimes_\pi L_q$ have finite cotype for $p \in (1, 2)$ and $q \in (1, 2]$?

## References

1. J. Briët, A. Naor and O. Regev, [Locally decodable codes and the failure of cotype for projective tensor products](https://doi.org/10.3934/era.2012.19.120), Electron. Res. Announc. Math. Sci. **19** (2012), 120–130; [arXiv:1208.0539](https://arxiv.org/abs/1208.0539).
2. O. Giladi, J. Prochno, C. Schütt, N. Tomczak-Jaegermann and E. Werner, [On the geometry of projective tensor products](https://doi.org/10.1016/j.jfa.2017.03.019), J. Funct. Anal. **273** (2017), 471–495.

## Source

Imported from [AIM Problem Lists convexbanach 2.1](http://aimpl.org/convexbanach/2/), curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.