Geometry of 2-homogeneous polynomials on \(\ell _{p}\) spaces, 1\(< p < \infty\)
type: Item
instance of: scholarly article
title: Geometry of 2-homogeneous polynomials on \(\ell _{p}\) spaces, 1\(< p < \infty\)
zbMATH Open document ID: 1024.46008
DOI: 10.1016/S0022-247X(02)00217-2
author: Bogdan C. Grecu
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The author considers spaces of 2-homogeneous continuous polynomials on the real space \(\ell_p\) (\({\mathcal P}(^2\ell_p)\)) and the real 2-dimensional space \(\ell_p^2\) (\({\mathcal P}(^2\ell_p^2)\)) for \(1<p<\infty\). Firstly, 2-homogeneous polynomials of diagonal form on \(\ell_p\) for \(p>2\) are characterized as extreme points of the unit ball of \({\mathcal P}(^2\ell_p)\). Secondly, the extreme points and smooth points of the unit ball of \({\mathcal P}(^2\ell_p^2)\) are described in the cases \(p>2\) and \(p<2\). Using the fact that \({\mathcal P}(^2\ell_p^2)\) is isometrically isomorphic to the dual of the symmetric projective tensor product \(\ell_p^2 \otimes_{s,\pi} \ell_p^2\), the author applies these results to study the geometry of this tensor product. Moreover, he shows that an extreme 2-homogeneous polynomial on \(\ell_p^2\) remains an extreme point if it is regarded as a polynomial on \(\ell_p\) for \(p>2\). The paper contains all proofs. Unfortunately, not all cases are considered in the proof of the first proposition.
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B04
zbMATH DE Number: 1866556
zbMATH Keywords: extreme points
zbMATH Keywords: smooth points
zbMATH Keywords: 2-homogeneous polynomials
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
cites work: Geometric theory of spaces of integral polynomials and symmetric tensor products
cites work: Extreme polynomials and multilinear forms on \(\ell_1\)
cites work: The unit ball of \({\mathcal P}(^2\ell_2^2)\)
cites work: Q4023934
cites work: Q4257163
cites work: Norm-attaining polynomials and differentiability
cites work: Q4952927
cites work: Geometry of three-homogeneous polynomials on real Hilbert spaces
cites work: Geometry of spaces of polynomials
cites work: Q4376458
full work available at URL: s0022-247x(02)00217-2
OpenAlex ID: W1985464644
description: scientific article; zbMATH DE number 1866556