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Let V be a two-dimensional real symmetric space with unit ball having 8n extreme points. Let λ(V) denote the absolute projection constant of V. We show that $\lambda \left(V\right)\le \lambda \left(V_n\right)$ where $V_n$ is the space whose ball is a regular 8n-polygon. Also we reprove a result of [1] and [5] which states that $4/\pi =\lambda \left(l{₂}^{\left(2\right)}\right)\ge \lambda \left(V\right)$ for any two-dimensional real symmetric space V.

We construct k-dimensional (k ≥ 3) subspaces $V^k$ of $l_1$, with a very simple structure and with projection constant satisfying $\lambda \left(V^k\right)\ge \lambda (V^k,l_1)>\lambda \left(l_2^{\left(k\right)}\right)$.

Let V be an n-dimensional real Banach space and let λ(V) denote its absolute projection constant. For any N ∈ N with N ≥ n define $\lambda {ₙ}^N=sup\lambda \left(V\right):dim\left(V\right)=n,V\subset l_{\infty }^{\left(N\right)}$, λₙ = supλ(V): dim(V) = n. A well-known Grünbaum conjecture [Trans. Amer. Math. Soc. 95 (1960)] says that λ₂ = 4/3. König and Tomczak-Jaegermann [J. Funct. Anal. 119 (1994)] made an attempt to prove this conjecture. Unfortunately, their Proposition 3.1, used in the proof, is incorrect. In this paper a complete proof of the Grünbaum conjecture is presented