Steven Heilman
ABSTRACT
It is shown that every measurable partition {A1,..., Ak} of R3 satisfies: ∑i=1k|intAi xe-1/2|x|22dx|22≤ 9π2. Let P1,P2,P3 be the partition of R2 into 120o sectors centered at the origin. The bound (1) is sharp, with equality holding if Ai=Pi x R for i∈ {1,2,3} and Ai=∅ 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.
Supplemental Material
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Index Terms
- Solution of the propeller conjecture in R3
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