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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 $\mathfrak{L}\left({}^n{l}_1^2\right)$ .

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

Norm(T) = {(x ₁, … , x _n) ∈ Eⁿ ∶ (x ₁, … , x _n) is a norming point of T}.

Norm(T) is called the norming set of T. We classify Norm(T) for every T ∈ $\mathfrak{L}$ (² 𝑑 _∗(1, w)²), where 𝑑 _∗(1, w)² = ℝ² with the octagonal norm of weight 0 < w < 1 endowed with ${‖\left(x,y\right)‖}_{d_{*}\left(1,w\right)}=\max \left\{\left|x\right|,\left|y\right|,\frac{\left|x\right|+\left|y\right|}{1+w}\right\}$ .

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 $$P{(}^2{ℝ}_{h(\frac{1}{2})}^2)$$.

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 [31].

Let 𝑛 ≥ 2. A continuous 𝑛-linear form 𝑇 on a Banach space 𝐸 is called norm-peak if there is a unique (𝑥₁, … , 𝑥_𝑛) ∈ 𝐸^𝑛 such that ║𝑥₁║ = … = ║𝑥_𝑛║ = 1 and for the multilinear operator norm it holds ‖𝑇 ‖ = |𝑇 (𝑥₁, … , 𝑥_𝑛)|.

Let 0 ≤ 𝜃 ≤ $\frac{\pi }{2}\text{ and }{\mathcal{l}}_{\left(\infty ,\theta \right)}^2$ = ℝ² with the rotated supremum norm ‖(𝑥, 𝑦)‖_(∞,𝜃) = max {|𝑥 cos 𝜃 + 𝑦 sin 𝜃|, |𝑥 sin 𝜃 − 𝑦 cos 𝜃|}.

In this note, we characterize all norm-peak multilinear forms on ${\mathcal{l}}_{\left(\infty ,\theta \right)}^2$ . As a corollary we characterize all norm-peak multilinear forms on ${\mathcal{l}}_p^2$ = ℝ² with the 𝓁_𝑝-norm for 𝑝 = 1, ∞.