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Let n ∈ ℕ. An element (x 1, … , xn ) ∈ En is called a norming point of T L n E if x 1 = = x n = 1 and T x 1 , , x n = T , where L n E denotes the space of all continuous symmetric n-linear forms on E. For T L n E , we define

Norm T = x 1 , , x n E n : x 1 , , x n  is a norming of  T .

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

Let · 2 be the plane with a certain norm such that the set of the extreme points of its unit ball ext B · 2 = ± W 1 , ± W 2 for some W 1 ± W 2 · 2 .

In this paper, we classify Norm(T) for every T L n · 2 . We also present relations between the norming sets of L n l 2 and L n l 1 2 .

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Let 𝑛 ∈ ℕ. An element (x 1, … , x 𝑛) ∈ En is called a norming point of T L ( nE) if ‖x 1‖ = ⋯ = ‖xn ‖ = 1 and |T (x 1, … , xn )| = ‖T‖, where L ( nE) denotes the space of all continuous n-linear forms on E. For T 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 L (2 𝑑 (1, w)2), where 𝑑 (1, w)2 = ℝ2 with the octagonal norm of weight 0 < w < 1 endowed with x , y d * 1 , w = max x , y , x + y 1 + w .

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

Let 0 ≤ 𝜃 ≤ π 2  and   l , θ 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 l , θ 2 . As a corollary we characterize all norm-peak multilinear forms on l p 2 = ℝ2 with the 𝓁𝑝-norm for 𝑝 = 1, ∞.

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We classify the extreme 2-homogeneous polynomials on 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 h ( 1 2 ) 2 ) .

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Abstract

For n,m≥ 2 this paper is devoted to the description of the sets of extreme and exposed points of the closed unit balls of ( l n m ) and s ( l n m ) , where ( l n m ) is the space of n-linear forms on m with the supremum norm, and s ( l n m ) is the subspace of ( l n m ) consisting of symmetric n-linear forms. First we classify the extreme points of the unit balls of ( l n m ) and s ( l n m ) , respectively. We show that ext B ( l n m ) ⊂ ext B ( l n m + 1 ) , which answers the question in [32]. We show that every extreme point of the unit balls of ( l n m ) and s ( l n m ) is exposed, correspondingly. We also show that
ext B s ( l n 2 ) = ext  B ( l n 2 ) s ( l n 2 ) ,
ext  B s ( l 2 m + 1 ) ext  B ( l 2 m + 1 ) s ( l 2 m + 1 ) ,
exp B S ( l n 2 ) = exp B ( l n 2 ) s ( l n 2 )

and exp B s ( l 2 m + 1 ) exp B ( l 2 m + 1 ) s ( l 2 m + 1 ) ,

which answers the questions in [31].

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