An Upper Bound on Grothendieck's Constant
This work provides a tighter theoretical bound on a fundamental constant in functional analysis and optimization, resolving a long-standing open problem for the mathematics community.
The authors improve the upper bound on Grothendieck's real constant, resolving a 2011 conjecture by Braverman et al. They achieve a bound of K_G < π/(2 log(1+√2)) - 10^{-217} using degree-three Hermite polynomials, and further tighten it to K_G < π/(2 log(1+√2)) - 10^{-5} via computer-assisted proof.
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.