Solution of the Propeller Conjecture in ℝ3

Steven Heilman, Aukosh Jagannath, Assaf Naor

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)263-305
Number of pages43
JournalDiscrete and Computational Geometry
Volume50
Issue number2
DOIs
StatePublished - Sep 2013

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Propellers
Hardness
Partition
Pi
Finite Set
Equality
Sector
Clustering
Game
kernel

ASJC Scopus subject areas

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

Cite this

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

In: Discrete and Computational Geometry, Vol. 50, No. 2, 09.2013, p. 263-305.

Research output: Contribution to journalArticle

Heilman, Steven ; Jagannath, Aukosh ; Naor, Assaf. / Solution of the Propeller Conjecture in ℝ3 . In: Discrete and Computational Geometry. 2013 ; Vol. 50, No. 2. pp. 263-305.
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