The Grothendieck constant is strictly smaller than Krivine's bound
  We prove that $K_G<\frac{\pi}{2\log(1+\sqrt{2})}$, where $K_G$ is the Grothendieck constant. 
Mark Braverman, Konstantin Makarychev, Yury Makarychev, Assaf Naor
math.FA cs.DS