On the topology of some hyperspaces of convex bodies associated to tensor norms

For every tuple $d_1,\dots, d_l\geq 2,$ let $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}$ denote the tensor product of $\mathbb{R}^{d_i},$ $i=1,\dots,l.$ Let us by $\mathcal{B}(d)$ hyperspace centrally symmetric convex bodies in $\mathbb{R}^d,$ $d=d_1\cdots d_l,$ endowed with Hausdorff distance, and $\mathcal{B}_\otimes(d_1,\dots,d_l)$ subset consisting that are closed unit balls reasonable crossnorms on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}.$ It is known a closed, contractible locally compact $\mathcal{B}(d).$ The called space tensorial bodies. In this work we determine homeomorphism type $\mathcal{B}_\otimes(d_1,\dots,d_l).$ We show even if not respect to Minkowski sum, it an absolute retract homeomorphic $\mathcal{Q}\times\mathbb{R}^p,$ where $\mathcal{Q}$ Hilbert cube $p=\frac{d_1(d_1+1)+\cdots+d_l(d_l+1)}{2}.$ Among other results, relation between Banach-Mazur compactum associated examined.