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Mathematics & Statistics

Colloquia — Summer 2008

Friday, May 9, 2008



On the Grunbaum “\(4/3\)” Conjecture
Grzegorz Lewicki
Jagiellonian University
PHY 130
Lesław Skrzypek


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 $$ \lambda_n=\sup\{\lambda(V):\operatorname{dim}(V)=n\}. $$ 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. König and N. T. Jaegermann, Norms of minimal projections, J. Funct. Anal. 119 (1994), 253-280.