Solution of the Propeller Conjecture in ℝ³

It is shown that every measurable partition {A₁,...,A_k} 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 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.