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

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 - 2012 |

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

### Other

Other | 44th Annual ACM Symposium on Theory of Computing, STOC '12 |
---|---|

Country | United States |

City | New York, NY |

Period | 5/19/12 → 5/22/12 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

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

**Solution of the propeller conjecture in ℝ 3 .** / Heilman, Steven; Jagannath, Aukosh; Naor, Assaf.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing.*pp. 269-276, 44th Annual ACM Symposium on Theory of Computing, STOC '12, New York, NY, United States, 5/19/12. https://doi.org/10.1145/2213977.2214003

}

TY - GEN

T1 - Solution of the propeller conjecture in ℝ 3

AU - Heilman, Steven

AU - Jagannath, Aukosh

AU - Naor, Assaf

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - grothendieck inequalities

KW - kernel clustering

KW - semidefinite programming

KW - unique games hardness

UR - http://www.scopus.com/inward/record.url?scp=84862635715&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862635715&partnerID=8YFLogxK

U2 - 10.1145/2213977.2214003

DO - 10.1145/2213977.2214003

M3 - Conference contribution

AN - SCOPUS:84862635715

SN - 9781450312455

SP - 269

EP - 276

BT - STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing

ER -