A numerical proof of the Grünbaum conjecture

The Hahn-Banach theorem states that onto each line in every normed space, there is a unitary projection, and Kadec and Snobar [KS71] proved (using John’s ellipsoid) that onto each n-dimensional subspace of any real normed space, there is a projection with norm at most λn 6 √ n . Grünbaum [Grü60] conjectured that λ2 = 4/3 = 1.333... < 1.414... = √ 2 , which is the projection constant of the plane of equation x1 + x2 + x3 = 0 in (R 3 , ‖ ‖∞) whose norm is hexagonal, hence the 4/3... Several attempts have been made to prove this conjecture: König and Tomczak-Jaegermann published in [KTJ94] a proof that was shown incomplete by Chalmers and Lewicki, who gave their own (a bit intricate) proof in [CL10]. Here is a simpler proof, mostly based on their works, and partially on a few numerical studies of extrema of functions of 3 variables. Using arguments due to Lewis [Lew88], König and Tomczak-Jaegermann proved that if On denotes the space of n orthonormal vectors in the standard Euclidean space R for every integers 1 6 n 6 N and if we set for all ( u , (x , y)) ∈ O1 × O2 : Φ2 (u , (x , y)) := ∑