The König constant is one

Abstract

For each $N\ge 1$, consider the normalized König bilinear form $B_{\mathrm{K}}:L_{\infty }({\mathbb{R}}^N)\times L_{\infty }({\mathbb{R}}^N)\to \mathbb{R}$ given by \[ B_{\mathrm K}(f,g):=\frac{1}{(\sqrt{2}π)^N} \iint_{\mathbb R^N\times\mathbb R^N} f(x)g(y)e^{-(\lVert x\rVert^2+\lVert y\rVert^2)/2} \sin\langle x,y\rangle\,\mathrm d x\,\mathrm d y, \] We define the König constant by \[ \mathfrak K_{\mathrm K}:=\sup_{N\geq1}\sup_{\substack{f,g:\mathbb R^N\to\{\pm1\}\\ f,g\ \mathrm{measurable}}}B_{\mathrm K}(f,g). \] The study of this bilinear form arose from efforts to determine the exact value of the Grothendieck constant. König~\cite{KONIG} conjectured that the sharp value should instead be given by the one-dimensional half-spaces $B_{\mathrm{K}}(\mathrm{sgn}(x_1),\mathrm{sgn}(x_1))=\frac{2}{\pi }\log (1+\sqrt{2})$. A positive answer to this conjecture, together with a classical upper bound of Krivine \cite{KRIVINE}, would determine the exact value of the Grothendieck constant. In a breakthrough~\cite{BMMN}, Braverman, Makarychev, Makarychev, and Naor disproved König's conjecture already in dimension two and used their counterexamples to obtain the first strict improvement over Krivine's bound. One question in \cite{BMMN} attempts to determine the Grothendieck constant through alternating Krivine rounding schemes arising from König's bilinear form in high dimension. More recently, Li et al.~\cite{LISK} constructed high-dimensional examples showing that ${\mathfrak{K}}_{\mathrm{K}}\ge 0.59357$. An elementary Fourier argument gives ${\mathfrak{K}}_{\mathrm{K}}\le 1$ and excludes equality for every finite-dimension. In this paper, we prove that ${\mathfrak{K}}_{\mathrm{K}}=1$ by constructing a family of Boolean pairs in high dimensions. In particular, this gives a negative answer to the high-dimensional aspect of the question in \cite{BMMN}.

22 pages