By a result of Bohnenblust for every three-dimensional normed space
\(X\) and its two-dimensional subspace \(Y\), there exists a
projection \(P\colon X \to Y\) such that \(\|P\| \leq
\frac{4}{3}\). The aim of the talk is to give a sketch of the proof
of the following theorem: if for some subspace \(Y\) the minimal
projection \(P\colon X \to Y\) satisfies \(\|P\| \geq \frac{4}{3}-R\)
for some \(R>0\), then there exists two dimensional subspace \(Z\)
of \(X\) and projection \(Q\colon X \to Z\) for which \(\|Q\| \leq 1
+ \Phi(R)\) where \(\Phi(R) \to 0\) as \(R \to 0\). In other words,
every space which has a subspace of almost maximal projection
constant has also a subspace of almost minimal projection
constant. As a consequence, every three-dimensional space has a
subspace with the projection constant strictly less than
\(\frac{4}{3}\), which gives a non-trivial upper bound for the
problem posed by Bosznay and Garay. We shall also characterize all
three-dimensional spaces which have a subspace with the projection
constant equal to \(\frac{4}{3}\).
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