Grothendieck’s inequality

Let \(\{x_{i}\}_{i=1}^{m}, \{y_{j}\}_{j=1}^{n} \subseteq S^{d-1}\) be two sets of points on the \((d-1)\)-sphere. Take \(A = [a_{ij}] \in \mathbb{R}^{n\times m}\) a fixed real matrix.

We want a constant \(K\) such that \[\begin{equation} \max\left\{ \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} \langle x_{i}, y_{j}\rangle \mid \{x_{i}\}_{i=1}^{m}, \{y_{j}\}_{j=1}^{n} \subseteq S^{d-1} \right\} \leq K \cdot \max\left\{ \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} \delta_{i} \varepsilon_{j} \mid \{\delta_{i}\}_{i=1}^{m}, \{\varepsilon_{j}\}_{j=1}^{n} \subseteq \{-1,1\} \right\} \end{equation}\] Define \[\begin{equation} K_{G} = \text{ smallest possible value of }K \end{equation}\]

Grothendieck’s inequality says that for every \(n,m \in \mathbb{N}\) and for any \(A\), there exists a universal value of \(K\) such that the above inequality is true. It has even an explicit form. One can have \(K_{G}\) between \[\begin{equation} \tfrac{\pi}{2} \exp(\eta_{0}^{2})\leq K_{G} \leq \frac{\pi}{2 \log( 1 + \sqrt{2})} = 1.78\dots, \end{equation}\] where \[\begin{equation} \eta_{0} = 0.25573 \dots \end{equation}\] is the unique solution of \[\begin{equation} 1 - \tfrac{2}{\pi} \int_{0}^{\eta_{0}} \exp(-\frac{z^{2}}{2}) \mathrm{d}z = \tfrac{2}{ \pi }e^{\eta_{0}^{2}} \end{equation}\]

This paper has a nice survey.