### 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) |
---|---|

Pages (from-to) | 263-305 |

Number of pages | 43 |

Journal | Discrete and Computational Geometry |

Volume | 50 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2013 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

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

**Solution of the Propeller Conjecture in ℝ3
.** / Heilman, Steven; Jagannath, Aukosh; Naor, Assaf.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 50, no. 2, pp. 263-305. https://doi.org/10.1007/s00454-013-9530-0

}

TY - JOUR

T1 - Solution of the Propeller Conjecture in ℝ3

AU - Heilman, Steven

AU - Jagannath, Aukosh

AU - Naor, Assaf

PY - 2013/9

Y1 - 2013/9

N2 - It is shown that every measurable partition {A1,...,Ak} ℝ3 satisfies, Let {P1, P2, P3} be the partition of ℝ2 into 120° sectors centered at the origin. The bound (1) is sharp, with equality holding if Ai = Pi × ℝ for i ∈ {1,2,3} and Ai = ∅ 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.

AB - It is shown that every measurable partition {A1,...,Ak} ℝ3 satisfies, Let {P1, P2, P3} be the partition of ℝ2 into 120° sectors centered at the origin. The bound (1) is sharp, with equality holding if Ai = Pi × ℝ for i ∈ {1,2,3} and Ai = ∅ 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.

UR - http://www.scopus.com/inward/record.url?scp=84881614344&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84881614344&partnerID=8YFLogxK

U2 - 10.1007/s00454-013-9530-0

DO - 10.1007/s00454-013-9530-0

M3 - Article

VL - 50

SP - 263

EP - 305

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -