An Upper Bound on Grothendieck's Constant

2026-05-29

AI Summary

This work addresses the problem of establishing a tight upper bound for Grothendieck’s real constant $K_G$, aiming to verify a conjecture proposed by Braverman et al. (2011). The authors introduce a novel approximation algorithm that eliminates the need for additional randomization by projecting vectors onto random planes through the origin and applying a rounding scheme based on fifth-order Hermite polynomials. Combining random projection, Hermite polynomial thresholding, interval arithmetic, and computer-assisted proof techniques, this approach provides the first confirmation of the conjecture without extra randomization. The method yields significantly improved bounds: theoretically, it establishes $K_G < \frac{\pi}{2 \log(1+\sqrt{2})} - 10^{-217}$ using third-order polynomials, and through rigorous numerical verification, it achieves a more practical bound of $K_G < \frac{\pi}{2 \log(1+\sqrt{2})} - 10^{-5}$.

Abstract

We show that Grothendieck's real constant $K_G$ can be upper bounded by projecting vectors onto a random plane through the origin and thresholding a degree five Hermite polynomial. This resolves a conjecture of Braverman-Makarychev-Makarychev-Naor from 2011, who required an extra randomization step in their rounding scheme and proved $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-500}$. As a corollary of our result, we prove the bound $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-217}$ by thresholding degree three Hermite polynomials in the plane. We finally give a rigorous computer-assisted proof that $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-5}$ using interval arithmetic and degree three Hermite polynomial thresholding.

Problem

• Grothendieck's constant
• upper bound
• Hermite polynomial
• rounding scheme
• interval arithmetic

Innovation

• Grothendieck constant
• Hermite polynomial thresholding
• random projection
• computer-assisted proof
• interval arithmetic