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On the geometry of projective tensor products

Banff International Research Station for Mathematical Innovation and Discovery

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 $1\leq p_1 \leq \dots\leq p_k \leq \infty$. Joint work with Giladi, Prochno, Tomczak-Jaegermann and Werner.

Mathematics; Convex and discrete geometry; Geometry; Discrete mathematics

video/mp4

Author affiliation: Christian-Albrechts-Universitaet

2017-11-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63690

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/