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 $4/\pi =\lambda \left(l{\u2082}^{\left(2\right)}\right)\ge \lambda \left(V\right)$ 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
$\lambda {\u2099}^{N}=sup\lambda \left(V\right):dim\left(V\right)=n,V\subset {l}_{\infty}^{\left(N\right)}$,
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

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