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A Lower Bound for Grothendieck's Constant

Steven Heilman

Abstract

We show that Grothendieck's real constant $K_{G}$ satisfies $K_G\geq c+10^{-26}$, improving on the lower bound of $c=1.676956674215576\ldots$ of Davie and Reeds from 1984 and 1991, respectively.

A Lower Bound for Grothendieck's Constant

Abstract

We show that Grothendieck's real constant satisfies , improving on the lower bound of of Davie and Reeds from 1984 and 1991, respectively.
Paper Structure (19 sections, 23 theorems, 209 equations)

This paper contains 19 sections, 23 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Upper Bounds on KG
  3. Lower Bounds on KG
  4. Brief Review of the Existing Lower Bound
  5. Our Contribution
  6. Technical Challenges
  7. Organization
  8. Conclusions and Future Directions
  9. Remark on the Complex Case
  10. Summary of Notation
  11. Review of Reeds
  12. Third Moment Bounds
  13. Maximizers have absolute value 1 a.e.
  14. Derivative Bound
  15. Fixed Neighborhood with Norm Drop
  16. ...and 4 more sections

Key Result

Theorem 1.1

Theorems & Definitions (45)

  • Theorem 1.1: Main
  • Lemma 2.1: Characterization of Maximizing Profiles
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1: Every maximizer of $\|R_\lambda\|_{\infty\to1}$ is $\{\pm1\}$-valued
  • proof
  • ...and 35 more