### Abstract

It is shown that every measurable partition {A _{1},..., A _{k}} of ℝ ^{3} satisfies (Equation Presented) Let {P _{1},P _{2},P _{3}} be the partition of ℝ ^{2} into 120° sectors centered at the origin. The bound (1) is sharp, with equality holding if A _{i}=P _{i} x ℝ for i ∈ {1,2,3} and A _{i} = ∅ for i∈ {4,...,k}. This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with 4 x 4 centered and spherical hypothesis matrix equals 2π/3.

Original language | English (US) |
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Title of host publication | STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing |

Pages | 269-276 |

Number of pages | 8 |

DOIs | |

State | Published - Jun 26 2012 |

Event | 44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States Duration: May 19 2012 → May 22 2012 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Other

Other | 44th Annual ACM Symposium on Theory of Computing, STOC '12 |
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Country | United States |

City | New York, NY |

Period | 5/19/12 → 5/22/12 |

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### Keywords

- grothendieck inequalities
- kernel clustering
- semidefinite programming
- unique games hardness

### ASJC Scopus subject areas

- Software

### Cite this

^{3}In

*STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing*(pp. 269-276). (Proceedings of the Annual ACM Symposium on Theory of Computing). https://doi.org/10.1145/2213977.2214003