
Journal of Convex Analysis 26 (2019), No. 4, 12971320 Copyright Heldermann Verlag 2019 Convex Bodies Associated to Tensor Norms Maite FernándezUnzueta Centro de Investigación en Matemáticas, A.P. 402 Guanajuato, Mexico maite@cimat.mx Luisa F. HiguerasMontano Centro de Investigación en Matemáticas, A.P. 402 Guanajuato, Mexico fher@cimat.mx [Abstractpdf] We determine when a convex body in $\mathbb{R}^d$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots \otimes\mathbb{R}^{d_l},$ $d=d_1\cdots d_l.$ We call these convex bodies ``tensorial bodies''. We prove that, among them, the only ellipsoids are the closed unit balls of Hilbert tensor products of Euclidean spaces. It is also proved that linear isomorphisms on $\mathbb{R}^{d_1}\otimes\cdots \otimes \mathbb{R}^{d_l}$ preserving decomposable vectors map tensorial bodies into tensorial bodies. This leads us to define a BanachMazur type distance between them, and to prove that there exists a BanachMazur type compactum of tensorial bodies. Keywords: Convex body, tensor norm, Minkowski space, BanachMazur distance, tensor product of convex sets, linear mappings on tensor spaces. MSC: 46M05, 52A21, 46N10, 15A69 [ Fulltextpdf (190 KB)] for subscribers only. 