Solution of the propeller conjecture in ℝ 3

Steven Heilman; Aukosh Jagannath; Assaf Naor

doi:10.1145/2213977.2214003

Solution of the propeller conjecture in ℝ ³

Steven Heilman, Aukosh Jagannath, Assaf Naor

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

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	https://doi.org/10.1145/2213977.2214003
State	Published - 2012
Externally published	Yes
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
ISSN (Print)	0737-8017

Other

Other	44th Annual ACM Symposium on Theory of Computing, STOC '12
Country/Territory	United States
City	New York, NY
Period	5/19/12 → 5/22/12

All Science Journal Classification (ASJC) codes

Software

Keywords

grothendieck inequalities
kernel clustering
semidefinite programming
unique games hardness

Access to Document

10.1145/2213977.2214003

Cite this

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title = "Solution of the propeller conjecture in ℝ 3 ",

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.",

keywords = "grothendieck inequalities, kernel clustering, semidefinite programming, unique games hardness",

author = "Steven Heilman and Aukosh Jagannath and Assaf Naor",

year = "2012",

doi = "10.1145/2213977.2214003",

language = "English (US)",

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series = "Proceedings of the Annual ACM Symposium on Theory of Computing",

pages = "269--276",

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note = "44th Annual ACM Symposium on Theory of Computing, STOC '12 ; Conference date: 19-05-2012 Through 22-05-2012",

}

Heilman, S, Jagannath, A & Naor, A 2012, Solution of the propeller conjecture in ℝ ³ in STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing. Proceedings of the Annual 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

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AU - Jagannath, Aukosh

AU - Naor, Assaf

PY - 2012

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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.

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