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

For every tuple $d_1,dots, d_lgeq 2,$ let $mathbb{R}^{d_1}otimescdotsotimesmathbb{R}^{d_l}$ denote the tensor product of $mathbb{R}^{d_i},$ $i=1,dots,l.$ Let us denote by $mathcal{B}(d)$ the hyperspace of centrally symmetric convex bodies in $mathbb{R}^d,$ $d=d_1cdots d_l,$ endowed with the Hausdorff distance, and by $mathcal{B}_otimes(d_1,dots,d_l)$ the subset of $mathcal{B}(d)$ consisting of the convex bodies that are closed unit balls of reasonable crossnorms on $mathbb{R}^{d_1}otimescdotsotimesmathbb{R}^{d_l}.$ It is known that $mathcal{B}_otimes(d_1,dots,d_l)$ is a closed, contractible and locally compact subset of $mathcal{B}(d).$ The hyperspace $mathcal{B}_otimes(d_1,dots,d_l)$ is called the space of tensorial bodies. In this work we determine the homeomorphism type of $mathcal{B}_otimes(d_1,dots,d_l).$ We show that even if $mathcal{B}_otimes(d_1,dots,d_l)$ is not convex with respect to the Minkowski sum, it is an Absolute Retract homeomorphic to $mathcal{Q}timesmathbb{R}^p,$ where $mathcal{Q}$ is the Hilbert cube and $p=frac{d_1(d_1+1)+cdots+d_l(d_l+1)}{2}.$ Among other results, the relation between the Banach-Mazur compactum and the Banach-Mazur type compactum associated to $mathcal{B}_otimes(d_1,dots,d_l)$ is examined.