标准单纯形和复数不是最稳定的噪声
Standard Simplices and Pluralities are Not the Most Noise Stable
摘要：
标准单纯形猜想和复数最稳定猜想是两个分别说明某些分区在高斯和离散噪声稳定性方面是最优的猜想。这两个定理是Borell（1985）的高斯噪声稳定性结果和多数是最稳定定理（2004）的自然推广。在这里，我们表明，标准的单纯形是不是最稳定的高斯空间中的分区和多元性是不是最稳定的低影响分区在离散空间中的每一个部分$k \geq 3$的数量，为每个值$ρ\neq 0$的噪声和每个规定的措施，只要他们不都等于$1/k$的不同部分。我们的结果并不矛盾的原始声明的多元性是稳定的和标准单纯形猜想在其原来的声明有关分区的集合的相等的措施。然而，他们指出，如果这些命题是真的，他们的准确性和他们的证明将至关重要地依赖于假设的集合是平等的措施，在形成鲜明对比，博雷尔的结果，多数是稳定定理和许多其他结果在等周理论。考虑到我们的结果，自然会要求（约束）分区实现最佳噪声稳定性。
英文摘要：
The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts $k \geq 3$, for every value $ρ\neq 0$ of the noise and for every prescribed measures for the different parts as long as they are not all equal to $1/k$. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell's result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.
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