Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solution of the propeller conjecture in $\mathbb{R}^3$   It is shown that every measurable partition ${A_1,..., A_k}$ of
$\mathbb{R}^3$ satisfies $$\sum_{i=1}^k||\int_{A_i}
xe^{-\frac12||x||_2^2}dx||_2^2\le 9\pi^2.\qquad(*)$$ Let ${P_1,P_2,P_3}$ be the
partition of $\mathbb{R}^2$ into $120^\circ$ sectors centered at the origin.
The bound is sharp, with equality holding if $A_i=P_i\times \mathbb{R}$ for
$i\in {1,2,3}$ and $A_i=\emptyset$ for $i\in \{4,...,k\}$ (up to measure zero
corrections, orthogonal transformations and renumbering of the sets
$\{A_1,...,A_k\}$). This settles positively the 3-dimensional Propeller
Conjecture of Khot and Naor (FOCS 2008). The proof of 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 $(*)$ is
complexity-theoretic: the Unique Games hardness threshold of the Kernel
Clustering problem with $4 \times 4$ centered and spherical hypothesis matrix
equals $\frac{2\pi}{3}$.

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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: math-ph
9
The categories of this paper are: nlin
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The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
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The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph