Geometry of 2-homogeneous polynomials on \(\ell _{p}\) spaces, 1\(< p < \infty\)
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.
scientific article; zbMATH DE number 1866556