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

Abstract

For every tuple d1,…,dl≥2, let Rd1⊗⋯⊗Rdl denote the tensor product of Rdi, i=1,…,l. Let us denote by B(d) the hyperspace of centrally symmetric convex bodies in Rd, d=d1⋯dl, endowed with the Hausdorff distance, and by B⊗(d1,…,dl) the subset of B(d) consisting of the convex bodies that are closed unit balls of reasonable crossnorms on Rd1⊗⋯⊗Rdl. It is known that B⊗(d1,…,dl) is a closed, contractible and locally compact subset of B(d). The hyperspace B⊗(d1,…,dl) is called the space of tensorial bodies. In this work we determine the homeomorphism type of B⊗(d1,…,dl). We show that even if B⊗(d1,…,dl) is not closed with respect to the Minkowski sum, it is an absolute retract homeomorphic to Q×Rp, where Q is the Hilbert cube and p=d1(d1+1)+⋯+dl(dl+1)2. Among other results, the relation between the Banach-Mazur compactum and the Banach-Mazur type compactum associated to B⊗(d1,…,dl) is examined.