Norm-Peak Multilinear Forms on the Plane with the Rotated Supremum Norm

Let 𝑛 ≥ 2. A continuous 𝑛-linear form 𝑇 on a Banach space 𝐸 is called norm-peak if there is a unique (𝑥 1 , … , 𝑥 𝑛 ) ∈ 𝐸 𝑛 such that ║𝑥 1 ║ = … = ║𝑥 𝑛 ║ = 1 and for the multilinear operator norm it holds ‖𝑇 ‖ = |𝑇 (𝑥 1 , … , 𝑥 𝑛 )|. Let 0 ≤ 𝜃 ≤ = ℝ 2 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 . As a corollary we characterize all norm-peak multilinear forms on = ℝ 2 with the 𝓁 𝑝 -norm for 𝑝 = 1, ∞.