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

Abstract

For every tuple $d_1,\dots ,d_l\ge 2,$ 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 closed 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.

28 pages. Among others, in this version we added an illustrative figure for the proof of Lemma 2.6. A gap on the selection of $r$ in the proof of lemma 2.6 was corrected. We provide a new sentence for Proposition 5.2. This new statement improves the result