# Mathematical Sciences Research Institute

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# Seminar

GFA Main Seminar: On the geometry of projective tensor products November 30, 2017 (11:00 AM PST - 12:00 PM PST)
Parent Program: Geometric Functional Analysis and Applications Space Science Lab, Room 105
Speaker(s) Carsten Schuett (Christian-Albrechts Universität Kiel)
Description No Description
Video
In this work, we study the volume ratio of the projective tensor products $\ell^n_p\otimes_{\pi}\ell_q^n\otimes_{\pi}\ell_r^n$ with $1\leq p\leq q \leq r \leq \infty$. The asymptotic formulas we obtain are sharp in almost all cases. As a consequence of our estimates, these spaces allow for an almost Euclidean decomposition of Kashin type whenever $1\leq p \leq q\leq r \leq 2$ or $1\leq p \leq 2 \leq r \leq \infty$ 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 the $k$-fold products $\ell_{p_1}^n\otimes_{\pi}\dots \otimes_{\pi}\ell_{p_k}^n$ with $k\in\N$ and $1\leq p_1 \leq \dots\leq p_k \leq \infty$.