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Abstract

and $\text{exp}{B}_{{ℒ}_s\left({}^{2}l{}_{\infty }^{m+1}\right)}\ne \text{exp}{B}_{ℒ\left({}^{2}l{}_{\infty }^{m+1}\right)}\cap {ℒ}_s\left({}^{2}l{}_{\infty }^{m+1}\right),$

which answers the questions in [].

We classify the extreme 2-homogeneous polynomials on $$ℝ$$ ² with the hexagonal norm of weight ½. As applications, using its extreme points with the Krein-Milman Theorem, we explicitly compute the polarization and unconditional constants of $$P{(}^2{ℝ}_{h(\frac{1}{2})}^2)$$.

Let n ∈ ℕ. An element (x ₁, … , x_n ) ∈ Eⁿ is called a norming point of $T\in \mathfrak{L}\left({}^{n}E\right)$ if $‖x_1‖=\cdots =‖x_n‖=1$ and $\left|T\left(x_1,\dots ,x_n\right)\right|=‖T‖$ , where $\mathfrak{L}\left({}^{n}E\right)$ denotes the space of all continuous symmetric n-linear forms on E. For $T\in \mathfrak{L}\left({}^{n}E\right)$ , we define

Norm(T) is called the norming set of T.

Let ${ℝ}_{‖·‖}^2$ be the plane with a certain norm such that the set of the extreme points of its unit ball ext $B_{{ℝ}_{‖·‖}^2}=\left\{\pm W_1,\pm W_2\right\}$ for some $W_1\ne \pm W_2\in {ℝ}_{‖·‖}^2$ .

In this paper, we classify Norm(T) for every $T\in \mathfrak{L}\left({}^n{ℝ}_{‖·‖}^2\right)$ . We also present relations between the norming sets of $\mathfrak{L}\left({}^n{l}_{\infty }^2\right)$ and $\mathfrak{L}\left({}^n{l}_1^2\right)$ .