@article{EJP3083,
author = {Steven Heilman},
title = {Euclidean partitions optimizing noise stability},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {19},
year = {2014},
keywords = {Standard simplex, plurality, optimization, MAX-k-CUT, Unique Games Conjecture},
abstract = {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).},
pages = {no. 99, 1-37},
issn = {1083-6489},
doi = {10.1214/EJP.v19-3083},
url = {http://ejp.ejpecp.org/article/view/3083}}