An Upper Bound on Grothendieck's Constant

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 $K_G<\frac{\ddot{I}}{€}2\log (1+\sqrt{2})-10^{-500}$. As a corollary of our result, we prove the bound $K_G<\frac{\ddot{I}}{€}2\log (1+\sqrt{2})-10^{-217}$ by thresholding degree three Hermite polynomials in the plane. We finally give a rigorous computer-assisted proof that $K_G<\frac{\ddot{I}}{€}2\log (1+\sqrt{2})-10^{-5}$ using interval arithmetic and degree three Hermite polynomial thresholding.

37 pages, 2 figures