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Mathematics & Statistics

Colloquia — Summer 2008

Friday, May 9, 2008

Title
Speaker

Time
Place

On the Grunbaum “$$4/3$$” Conjecture
Grzegorz Lewicki
Jagiellonian University
Poland
3:00pm-4:00pm
PHY 130
Lesław Skrzypek

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 $$\lambda_n=\sup\{\lambda(V):\operatorname{dim}(V)=n\}.$$ 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.