On the geometry of projective tensor products

TL;DR

This paper analyzes the volume ratios of projective tensor products of $oldsymbol{ ext{l}^n_p}$ spaces, providing sharp asymptotic formulas and implications for Euclidean decompositions and cotype properties.

Contribution

It offers new asymptotic formulas for volume ratios of tensor products and extends results to general k-fold products, revealing geometric and cotype characteristics.

Findings

Sharp asymptotic formulas for volume ratios in most cases

Nearly Euclidean decompositions for specific parameter ranges

Insights into cotype properties of tensor product spaces

Abstract

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 $1\leq p\leq q \leq r \leq \infty$. 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 $1\leq p \leq q\leq r \leq 2$ 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 $k$-fold products $\ell_{p_1}^n\otimes_{\pi}\dots \otimes_{\pi}\ell_{p_k}^n$ with $k\in\mathbb N$ and $1\leq p_1 \leq \dots\leq p_k \leq \infty$.