It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product $\ell_p\tp X$ fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then $\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3}$ does not have finite cotype. This is proved via a connection to the theory of locally decodable codes.

cotype, projective tensor product, locally decodable codes
Life Sciences (theme 5), Logistics (theme 3)
American Institute of Mathematical Sciences
Electronic Research Announcements in Mathematical Sciences
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Briët, J, Naor, A, & Regev, O. (2012). Locally decodable codes and the failure of cotype for projective tensor products. Electronic Research Announcements in Mathematical Sciences, 19, 120–130.