Euclidean Partitions Optimizing Noise Stability

The Standard Simplex Conjecture of Isaksson and Mossel asks for the partition $\{A_i{\}}_{i=1}^k$ of ${\mathbb{R}}^n$ into $k\le n+1$ pieces of equal Gaussian measure of optimal noise stability. That is, for $\rho >0$, we maximize $$\sum _{i=1}^k{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{1}_{A_i}(x)1_{A_i}(x\rho +y\sqrt{1-{\rho }^2})e^{-(x_1^2+\cdots +x_n^2)\mathrm{/}2}{e}^{-(y_1^2+\cdots +y_n^2)\mathrm{/}2}dxdy\mathrm{.}$$ Isaksson and Mossel guessed the best partition for this problem and proved some applications of their conjecture. For example, the Standard Simplex Conjecture implies the Plurality is Stablest Conjecture. For $k=3,n\ge 2$ and $0<\rho <{\rho }_0(k,n)$, we prove the Standard Simplex Conjecture. The full conjecture has applications to theoretical computer science, and to geometric multi-bubble problems (after Isaksson and Mossel).