A hyperspace of convex bodies arising from tensor norms

Abstract

In a preceding work it is determined when a convex body in Rd, d=d1⋯dl, is the unit ball of a reasonable crossnorm on Rd1⊗⋯⊗Rdl. Consequently, the class of tensorial bodies is introduced and a 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 Rd1⊗⋯⊗Rdl preserving decomposable vectors acts properly on it. A compact global slice for the space is constructed. Then, topological representatives for the space of tensorial bodies and the Banach-Mazur type compactum are given. It is showed that the set of ellipsoids in the class of tensorial bodies is homeomorphic to the Euclidean space of dimension p=d1(d1+1)2+⋯+dl(dl+1)2. We also prove the continuity of both the projective and the injective tensor product of 0-symmetric convex bodies, with respect to the Hausdorff distance.