The König constant is one

[2608.14817] The König constant is one Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Functional Analysis arXiv:2608.14817 (math) [Submitted on 14 Aug 2026] Title: The König constant is one Authors: Xinyuan Xie , Haonan Zhang View a PDF of the paper titled The K\"onig constant is one, by Xinyuan Xie and Haonan Zhang View PDF HTML (experimental) Abstract: For each $N\geq1$, 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 \[ B_{\mathrm K}(f,g):=\frac{1}{(\sqrt{2}\pi)^N} \iint_{\mathbb R^N\times\mathbb R^N} f(x)g(y)e^{-(\lVert x\rVert^2+\lVert y\rVert^2)/2} \sin\langle x,y\rangle\,\mathrm d x\,\mathrm d y, \] 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 $B_{\mathrm K}(\operatorname{sgn}(x_1),\operatorname{sgn}(x_1))=\frac{2}{\pi}\log(1+\sqrt{2})$. 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}. Comments: 22 pages Subjects: Functional Analysis (math.FA) ; Computational Complexity (cs.CC) Cite as: arXiv:2608.14817 [math.FA] (or arXiv:2608.14817v1 [math.FA] for this version) https://doi.org/10.48550/arXiv.2608.14817 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Haonan Zhang [ view email ] [v1] Fri, 14 Aug 2026 18:42:11 UTC (23 KB) Full-text links: Access Paper: View a PDF of the paper titled The K\"onig constant is one, by Xinyuan Xie and Haonan Zhang View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2026-08 Change to browse by: cs cs.CC math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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