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on The real Grothendieck constant ($C_{10a}$)

The statement

$C_{10}$ 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 $d$ ).

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.