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
$< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-5}$ [Hei26b] $10^{-5}$ improvement over Krivine’s bound
$< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 6.039\times 10^{-5}$ [LSXCKKM26] $6.039\times 10^{-5}$ improvement over Krivine’s bound
$< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-4}$ [SLXCKKM26] Same team; via the first asymptotic Krivine-type rounding scheme (previous works only considered low-dimensional schemes).

Known lower bounds

Bound Reference Comments
$1$ Trivial Follows from the definitions
$\dfrac{\pi}{2} \approx 1.57080$ [G1953] Grothendieck’s original lower bound
$K_{DR} \approx 1.67696\ldots$ [Dav1984], [Ree1991] Davie–Reeds lower bound
$K_{DR} + 10^{-26}$ [Hei26] Concurrent 2026 improvement over the Davie–Reeds bound
$K_{DR} + 10^{-12}$ [JM26] Strict improvement over the Davie–Reeds bound
$\dfrac{6\pi}{11}\approx 1.71360$ [SLXCKKM26] Major jump over the Davie–Reeds anchor $\approx 1.67696$; proved by establishing asymptotic limitations of Krivine schemes rather than by constructing explicit gap instances (as all prior lower bounds did). Together with the upper bound above, this determines the tenths digit of $K_G^{\mathbb R}$ to be $7$.

References