On the geometry of projective tensor products

In this work, we study the volume ratio of the projective tensor products ℓ_pⁿ⊗_πℓ_qⁿ⊗_πℓ_rⁿ with 1≤p≤q≤r≤∞. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kašin type whenever 1≤p≤q≤r≤2 or 1≤p≤2≤r≤∞ and q=2. Also, from the Bourgain–Milman bound on the volume ratio of Banach spaces in terms of their cotype 2 constant, we obtain information on the cotype of these 3-fold projective tensor products. Our results naturally generalize to k-fold products ℓ_p1ⁿ⊗_π…⊗_πℓ_pkⁿ with k∈N and 1≤p₁≤…≤p_k≤∞.