Journal of Convex Analysis 26 (2019), No. 4, 1297--1320
Copyright Heldermann Verlag 2019
Convex Bodies Associated to Tensor Norms
Maite Fernández-Unzueta
Centro de Investigación en Matemáticas, A.P. 402 Guanajuato, Mexico
maite@cimat.mx
Luisa F. Higueras-Montano
Centro de Investigación en Matemáticas, A.P. 402 Guanajuato, Mexico
fher@cimat.mx
[Abstract-pdf]
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 Banach-Mazur type distance between them, and to prove that there exists a Banach-Mazur type compactum of tensorial bodies.
Keywords: Convex body, tensor norm, Minkowski space, Banach-Mazur distance, tensor product of convex sets, linear mappings on tensor spaces.
MSC: 46M05, 52A21, 46N10, 15A69
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