Three-dimensional subspace of l∞(5) with maximal projection constant

Let V be an n  -dimensional real Banach space and let λ(V)λ(V) denote its absolute projection constant. For any N∈NN∈N, N⩾nN⩾n, defineλnN=sup{λ(V):dim(V)=n,V⊂l∞(N)} andλn=sup{λ(V):dim(V)=n}. A well-known Grünbaum conjecture (p. 465 in [B. Grünbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960) 451–465]) says thatλ2=4/3.λ2=4/3. In this paper we show thatλ35=5+427 and we determine a three-dimensional space V⊂l∞(5) satisfying λ35=λ(V). In particular, this shows that Proposition 3.1 from [H. König, N. Tomczak-Jaegermann, Norms of minimal projections, J. Funct. Anal. 119 (1994) 253–280] (see p. 259) is incorrect. Hence the proof of the Grünbaum conjecture given in [H. König, N. Tomczak-Jaegermann, Norms of minimal projections, J. Funct. Anal. 119 (1994) 253–280] which is based on Proposition 3.1 is incomplete.