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).
Highlights
For k = 3, n ≥ 2 and 0 < ρ < ρ0(k, n), we prove the Standard Simplex Conjecture
The Standard Simplex Conjecture [12] asks for the partition {Ai}ki=1 of Rn into k ≤ n + 1 sets of equal Gaussian measure of optimal noise stability
This Conjecture generalizes a seminal result of Borell, [3, 19], which corresponds to the k = 2 case of the Standard Simplex Conjecture
Summary
The Standard Simplex Conjecture [12] asks for the partition {Ai}ki=1 of Rn into k ≤ n + 1 sets of equal Gaussian measure of optimal noise stability. This approximation procedure, which uses an invariance principle, shows the equivalence of the Plurality is Stablest Conjecture and Standard Simplex Conjecture [12, Theorems 1.10 and 1.11]. By using this feedback loop, we show in Theorem 7.1 that a regular simplicial conical partition maximizes (d/dρ)J for small ρ > 0, k = 3, n ≥ 2. For small ρ < 0, (d/dρ)J is not maximized by the regular simplicial conical partition This result does not confirm or deny Conjecture 1 for ρ < 0.
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