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

TL;DR

This paper characterizes the topological structure of the space of convex bodies associated with tensor norms, showing it is homeomorphic to a product of a Hilbert cube and Euclidean space, revealing its geometric complexity.

Contribution

It determines the homeomorphism type of the hyperspace of tensorial convex bodies, establishing it as an absolute retract homeomorphic to a Hilbert cube times Euclidean space.

Findings

The space of tensorial convex bodies is an absolute retract.

It is homeomorphic to a product of a Hilbert cube and Euclidean space.

The relation between Banach-Mazur compactum and tensorial convex bodies is analyzed.

Abstract

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 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.