| Abstract |
Let $V$ be a real, finite-dimensional Banach space and let $\lambda;(V)$ denote its absolute projection constant. For any $n,N \in \mathbb{N}$, $n \leq N$ by $S_{n,N}$ we denote the set of all $n$-dimensional, real Banach spaces which can be isometrically embedded in $l_{\infty}^{(N)}$. Set
\lambda_n^N = \sup \{ \lambda(V): V \in S_{n,N}\},
and
\lambda_n = \sup \{ \lambda(V): dim(V) = n\}.
The famous Grunbaum conjecture [1] says that $\lambda_2 = 4/3$.
In my talk I will give a sketch of the proof of the fact that
\lambda_3^5 = \frac{5+4\sqrt{2}}{7}.
Also a three-dimensional space $V$ satisfying $\lambda(V) = \lambda_3^5$ will be determined. In particular, this shows that Proposition 3.1 from [2] is incorrect and consequently the proof of the Grunbaum conjecture presented in [2] is incomplete.
Next a sketch of a proof of Grunbaum's conjecture will be presented.
[1] B. Grunbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960), 451-465.
[2] H. Kőnig, N. T. Jaegermann, Norms of minimal projections, Journal of Functional Analysis 119 (1994), 253-280. |