Euclidean Partitions Optimizing Noise Stability

Abstract: The Standard Simplex Conjecture of Isaksson and Mossel asks for the partition ${A_{i}}<em>{i=1}<sup>{k}$ of $\mathbb{R}<sup>{n}$ into $k\leq n+1$ pieces of equal Gaussian measure of optimal noise stability. That is, for $\rho&gt;0$, we maximize $$\sum</em>{i=1}^{{k}\int_{\mathbb{R}^{{n}}\int_{\mathbb{R}^{{n}}1_{A_{i}}(x)1_{A_{i}}(x\rho+y\sqrt{1-\rho^{2}})}}} e^{{-(x_{1}^{{2}+\cdots+x_{n}^{{2})/2}e^{{-(y_{1}^{{2}+\cdots+y_{n}^{2})/2}dxdy.}}}}}$$ 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 $0&lt;\rho&lt;\rho_{0}(k,n)$, 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).