Cyclic group tensors and the volume ratio of projective tensor products

We revisit the problem of estimating the norm of the natural embedding from the Euclidean space ${\mathrm{\ell }}_2^{n^3}$ into the projective tensor product ${\mathrm{\ell }}_p^n{\otimes }_{\pi }{\mathrm{\ell }}_q^n{\otimes }_{\pi }{\mathrm{\ell }}_r^n$ in the range $1\le p\le q\le 2\le r\le \mathrm{\infty }$, studied by Giladi, Prochno, Schütt, Tomczak-Jaegermann and Werner [J. Funct. Anal. 273 (2017), 471-495]. Using convolution on finite cyclic groups, we establish new lower bounds that improve their estimates and determine the exact asymptotic order of the embedding norm whenever $1\mathrm{/}p+1\mathrm{/}q+1\mathrm{/}r\ge 2$. We then apply an estimate of Boedihardjo for the injective norm of random tensors to a randomly signed cyclic convolution tensor and use multilinear interpolation to determine the optimal exponent throughout the entire parameter range, up to logarithmic factors. As consequences, we obtain corresponding improvements and asymptotically sharp estimates, up to logarithmic factors, for the volume ratio of these projective tensor products.