  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 denote by $\mathcal{B}(d)$
the hyperspace of centrally symmetric convex bodies in $\mathbb{R}^d,$
$d=d_1\cdots d_l,$ 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
$\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}.$ 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
$\mathcal{Q}\times\mathbb{R}^p,$ 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.
