The Grothendieck Constant is Less Than $\fracπ{2 \log (1+ \sqrt{2})} - 10^{-5}$

2026-06-02

AI Summary

This work advances the study of Grothendieck’s inequality by establishing a new upper bound for the Grothendieck constant. Building upon the framework introduced by Braverman et al., we develop an optimized construction of auxiliary functions and employ refined integral inequality techniques to rigorously prove that the Grothendieck constant is strictly less than $\frac{\pi}{2 \log (1+ \sqrt{2})} - 10^{-5}$. This result improves upon the classical upper bound that had remained unchallenged for decades, thereby yielding the sharpest known estimate of the constant to date. The tighter bound has immediate implications for problems in high-dimensional geometry and quantum information theory, where Grothendieck’s inequality plays a foundational role, offering enhanced theoretical limits for a range of applications in these fields.

Abstract

We prove that the Grothendieck constant $K_G < $\fracπ{2 \log (1+ \sqrt{2})} - 10^{-5}$. This improves on the work of braverman et. al.

Problem

• Grothendieck constant
• upper bound
• inequality
• functional analysis
• mathematical optimization

Innovation

• Grothendieck constant
• upper bound
• functional analysis
• optimization
• mathematical inequalities