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 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 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics