Abstract

We determine when a convex body in Rd\mathbb{R}^d is the closed unit ball of a reasonable crossnorm on Rd1⊗⋯⊗Rdl,\mathbb{R}^{d_1}\otimes\cdots \otimes\mathbb{R}^{d_l}, d=d1⋯dl.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 Rd1⊗⋯⊗Rdl\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.

Author information

Contact details are reproduced from the original publication and may be historical.

Maite Fernández-Unzueta

Centro de Investigación en Matemáticas, A.P. 402 Guanajuato, México

maite@cimat.mx

Luisa F. Higueras-Montaño

Centro de Investigación en Matemáticas, A.P. 402 Guanajuato, México

fher@cimat.mx

Keywords

  • Convex body
  • tensor norm
  • Minkowski space
  • Banach-Mazur distance
  • tensor product of convex sets
  • linear mappings on tensor spaces

Mathematics Subject Classification

46M05, 52A21, 46N10, 15A69

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Suggested citation

M. Fernández-Unzueta, L. F. Higueras-Montaño. “Convex Bodies Associated to Tensor Norms.” Journal of Convex Analysis 26 (2019), No. 4, 1297–1320. https://doi.org/10.68381/jca26068

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press.