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2014 Euclidean partitions optimizing noise stability
Steven Heilman
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Electron. J. Probab. 19: 1-37 (2014). DOI: 10.1214/EJP.v19-3083


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).


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Steven Heilman. "Euclidean partitions optimizing noise stability." Electron. J. Probab. 19 1 - 37, 2014.


Accepted: 15 August 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1364.60010
MathSciNet: MR3256871
Digital Object Identifier: 10.1214/EJP.v19-3083

Primary: 68Q25

Keywords: MAX-k-CUT , optimization , plurality , Standard simplex , Unique Games Conjecture

Vol.19 • 2014
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