The real Grothendieck constant

Description of constant

$C_{10}$ is the real Grothendieck constant $K_{G}^{\mathbb R}$.

It is the smallest constant $C$ such that for every $m,n \ge 1$ and every real matrix

$A=(a_{ij}) \in \mathbb{R}^{m\times n}$ one has

\[\max_{\substack{u_1,\dots,u_{m}, v_{1},\dots,v_{n} \in S^{\infty}}} \ \sum_{i=1}^m \sum_{j=1}^n a_{ij} \langle u_{i}, v_{j}\rangle \ \le\ C \max_{\varepsilon_{1},\dots,\varepsilon_{m}, \delta_{1},\dots,\delta_{n} = \pm 1} \ \sum_{i=1}^m \sum_{j=1}^n a_{ij} \varepsilon_{i} \delta_{j}.\]

Here $S^{\infty}$ denotes the unit sphere of a real Hilbert space (equivalently, one may take $u_{i},v_{j} \in S^{d-1}$ for some sufficiently large finite dimension $d$).

Known upper bounds

Bound Reference Comments
$\sinh(\pi/2) \approx 2.30130$ [G1953] Grothendieck’s original upper bound
$2.261$ [R1974] Improvement of the original upper bound
$\dfrac{\pi}{2\ln(1+\sqrt{2})} \approx 1.782214$ [K1979] Krivine’s bound; best known explicit numerical upper bound
$< \dfrac{\pi}{2\ln(1+\sqrt{2})}$ [BMMN2011] Strict improvement over Krivine’s bound (no widely cited explicit numerical gap)

Known lower bounds

Bound Reference Comments
$1$ Trivial Follows from the definitions
$\dfrac{\pi}{2} \approx 1.57080$ [G1953] Grothendieck’s original lower bound
$1.67696$ [Dav1984], [Ree1991] Best known lower bound (due to Davie and independently Reeds)

References