### Abstract

It is shown that every measurable partition {A_{1},...,A_{k}} ℝ^{3} satisfies, 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} × 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 [(Mathematika 55(1-2):129-165, 2009 (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 × 4 centered and spherical hypothesis matrix equals 2Π/3.

Original language | English (US) |
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Pages (from-to) | 263-305 |

Number of pages | 43 |

Journal | Discrete and Computational Geometry |

Volume | 50 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2013 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

^{3}.

*Discrete and Computational Geometry*,

*50*(2), 263-305. https://doi.org/10.1007/s00454-013-9530-0