### Abstract

It is shown that for every k ∈ ℕ there exists a Borel probability measure μ on {−1, 1}^{ℝk} × {−1, 1}^{ℝk} such that for every m, n ∈ ℕ and x_{1},...,x_{m}, y_{1},..., y_{n} ∈ S^{m+n−1} there exist x′_{1},..., x′_{m}, y′_{1},..., y′_{n} ∈ S ^{m+n−1} such that if G : ℝ^{m+n} → ℝ^{k} is a random k × (m + n) matrix whose entries are i.i.d. standard Gaussian random variables, then for all (i, j) ∈ {1,...,m} × {1,...,n} we have where K_{G} is the real Grothendieck constant and C ∈ (0, ∞) is a universal constant. This establishes that Krivine’s rounding method yields an arbitrarily good approximation of K_{G}.

Original language | English (US) |
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Pages (from-to) | 4315-4320 |

Number of pages | 6 |

Journal | Proceedings of the American Mathematical Society |

Volume | 142 |

Issue number | 12 |

DOIs | |

State | Published - Jan 1 2014 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Naor, A., & Regev, O. (2014). Krivine schemes are optimal.

*Proceedings of the American Mathematical Society*,*142*(12), 4315-4320. https://doi.org/10.1090/S0002-9939-2014-12169-1