2016/08/30 by Giladi, Ohad, Prochno, Joscha, Schütt, Carsten +2
#46A32 #46B07 #46B28 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1608.08436
In this work, we study the volume ratio of the projective tensor products ℓnp⊗πℓqn⊗πℓrn 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 Kashin 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 ℓp1n⊗π… ⊗πℓpkn with k∈\mathbb N and 1≤ p1 ≤ …≤ pk ≤ ∞.