A hyperspace of convex bodies arising from tensor norms

TL;DR

This paper develops a topological framework for the space of tensorial convex bodies, exploring their structure, symmetries, and relationships with tensor products, extending the understanding of convex bodies in tensor spaces.

Contribution

It introduces the hyperspace of tensorial convex bodies, constructs a global slice, and characterizes the topology and symmetries of this space, including the homeomorphism of ellipsoids to Euclidean space.

Findings

The space of tensorial bodies admits a proper action by a symmetry group.

Ellipsoids within tensorial bodies form a space homeomorphic to Euclidean space.

Tensor product operations are continuous with respect to the Hausdorff distance.

Abstract

In a preceding work it is determined when a centrally symmetric convex body in $\mathbb{R}^d,$ $d=d_1\cdots d_l,$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}.$ Consequently, the class of tensorial bodies is introduced, an associated tensorial Banach-Mazur distance is defined and the corresponding Banach-Mazur type compactum is proved to exist. In this paper, we introduce the hyperspace of these convex bodies. We called "the space of tensorial bodies". It is proved that the group of linear isomorphisms on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}$ preserving decomposable vectors acts properly (in the sense of Palais) on it. A convenient compact global slice for the space is constructed. With it, topological representatives for the space of tensorial bodies and the Banach-Mazur type compactum are given. Among others, it is showed that the set of ellipsoids in the class of tensorial bodies is homeomorphic to the Euclidean space of dimension $p=\frac{d_1(d_1+1)}{2}+\cdots+\frac{d_l(d_l+1)}{2}.$ We also prove that both the projective and the injective tensor products of $0$-symmetric convex bodies are continuous functions with respect to the Hausdorff distance.