Volume estimates in spaces of homogeneous polynomials
type: Item
instance of: scholarly article
title: Volume estimates in spaces of homogeneous polynomials
zbMATH Open document ID: 1171.46033
author: Andreas Defant
author: Christopher Prengel
published in: Mathematische Zeitschrift
publication date: 2009-03-26T00:00:00Z
review text: The authors consider a symmetric Banach sequence space \(X\) which is either 2-convex and has nontrivial concavity or 2-concave and has nontrivial convexity, and use \(X_n\) to denote the \(n\)-dimensional slice of \(X\) spanned by the first \(n\) basis vectors. They obtain asymptotic upper and lower bounds for the volume of the unit ball of the spaces of \(m\)-homogeneous polynomials, \({\mathcal P}(^mX_n)\), and \(m\)-homogeneous nuclear polynomials, \({\mathcal P}_N(^mX_n)\), in terms of the fundamental function of the space. This allows them to give asymptotic upper and lower bounds for the volume of the unit ball of both the full and symmetric spaces of injective and projective \(m\)-tensors on \(X_n\) in terms of the volume of the unit ball of \(X_n\). Using these results, estimates are then obtained for the volume ratio (the ratio of the volume of the ball to the volume of its ellipsoid of maximal volume) of the unit balls and the projective constants of spaces of injective and projective tensors on \(X_n\) and for the Banach--Mazur distance from the spaces of injective and projective tensors on \(X_n\) to the Hilbert spaces of the corresponding dimension. Specific and elegant formulae for all these isometric invariants are obtained in the case in which \(X\) is \(\ell_p\) with \(1\leq p\leq\infty\).
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 46B04
zbMATH DE Number: 5533157
zbMATH Keywords: volume estimates
zbMATH Keywords: spaces of tensor products
zbMATH Keywords: isometric invariants
zbMATH Keywords: asymptotic estimates
MaRDI profile type: Publication
full work available at URL: s00209-008-0358-x
OpenAlex ID: W2089844308
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DOI: 10.1007/S00209-008-0358-X
description: scientific article; zbMATH DE number 5533157