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

Let ≥ 2. A continuous -linear form on a Banach space is called norm-peak if there is a unique ( 1 , … , ) ∈ such that ║ 1 ║ = … = ║ ║ = 1 and for the multilinear operator norm it holds ‖ ‖ = | ( 1 , … , )|. Let 0 ≤ ≤ = ℝ 2 with the rotated supremum norm ‖( , )‖ (∞, ) = max {| cos + sin |, | sin − cos |}. 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 -norm for = 1, ∞.