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 $k\leq n+1$ pieces of equal Gaussian measure of optimal noise stability. That is, for $\rho>0$, we maximize $$ \sum{i=1}^{k}\int{\mathbb{R}^{{n}}\int{\mathbb{R}{n}}1{A{i}}(x)1{A_{i}}(x\rho+y\sqrt{1-\rho{2}})} e^{{-(x{1}{2}+\cdots+x{n}{2})/2}e{-(y{1}{2}+\cdots+y{n}{2})/2}dxdy.} $$ 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 $k=3,n\geq2$ 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).