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

Let V be an n-dimensional real Banach space and let λ(V ) denote its absolute projection constant. For any N ∈ N, N n, define λNn = sup λ(V ): dim(V ) = n, V ⊂ l (N) ∞ and λn = 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 that λ2 = 4/3. In this paper we show that λ53 = 5+4√2 7and we determine a three-dimensional space V ⊂ l (5) ∞ satisfying λ53 = 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. © 2009 Elsevier Inc. All rights reserved