## Euclidean partitions optimizing noise stability.(English)Zbl 1364.60010

Summary: The Standard Simplex Conjecture of M. Isaksson and E. Mossel [Isr. J. Math. 189, 347–396 (2012; Zbl 1256.60017)] 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 $\sum_{i=1}^k\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}1_{A_i}(x)1_{A_i}(x\rho+y\sqrt{1-\rho^2})e^{-(x_1^2+\ldots +x_n^2)/2}e^{-(y_1^2+\ldots+y_n^2)/2}\,dx\,dy.$
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).

### MSC:

 60A10 Probabilistic measure theory 60G15 Gaussian processes 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q25 Analysis of algorithms and problem complexity

Zbl 1256.60017
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