Solution of the propeller conjecture in R^3

Steven Heilman, Aukosh Jagannath, Assaf Naor

Abstract:

It is shown that every measurable partition $\{A_1,\ldots, A_k\}$ of $\mathbb{R}^3$ satisfies
$$
\sum_{i=1}^{k}\left\|\int_{A_i} xe^{-\frac12\|x\|_2^2}dx\right\|_2^2\le 9\pi^2. (\ast)
$$
Let $\{P_1,P_2,P_3\}$ be the partition of $\mathbb{R}^2$ into $120^\circ$
sectors centered at the origin. The bound $(\ast)$ is sharp, with equality holding if $A_i=P_i\times \mathbb{R}$ for $i\in \{1,2,3\}$ and $A_i=\emptyset$ for $i\in \{4,\ldots,k\}$. 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 $(\ast)$ is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with $4\times 4$ centered and spherical hypothesis matrix equals $\frac{2\pi}{3}$.