Grothendieck's Constant

Let A be an $n\times n$ Real Square Matrix and let $x_i$ and $y_j$ be real numbers with $\vert x_i\vert, \vert y_i\vert<0$. Then Grothendieck showed that there exists a constant $K$ independent of both A and $n$ satisfying

$$\left\vert{\,\sum_{1\leq i,j\leq n} a_{ij}\left\langle{x_i,y_j}\right\rangle{}\,}\right\vert\leq K$$ (1)

in which the vectors $x_i$ and $y_j$ have a norm $<1$ in any Hilbert Space. The Grothendieck constant is the smallest Real Number for which this inequality has been proven. Krivine (1977) showed that

$$1.676\ldots \leq K_G\leq 1.782\ldots,$$ (2)

and has postulated that

$$K_G\equiv {\pi\over 2\ln(1+\sqrt{2}\,)} = 1.7822139\ldots.$$ (3)

It is related to Khintchine's Constant.

References

Krivine, J. L. ``Sur la constante de Grothendieck.'' C. R. A. S. 284, 8, 1977.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 42, 1983.