Maximal vectors in some Hilbert spaces

Let V be a norm closed subset of the unit sphere of a Hilbert space H that is stable under multiplication by scalars of absolute value 1. The inner radius r(V) of V is the largest r⩾0 such that {ξ∈H:‖ξ‖⩽r} is contained in the closed convex hull of V. In multipartite tensor products H=H1⊗⋯⊗HN which are arranged such that the dimensions nk=dimHk weakly increase with k and nN-1<∞, Arveson calculated the inner radius r(V) for the case nN⩾n1n2⋯nN-1 and gave the form of maximal vectors in [J. Funct. Anal. 256 (2009) 1476–1510]. In this paper, we consider the inner radius r(V) and describe the set of maximal vectors in the remaining cases for which nN≤n1⋯nN-1.