We introduce a new notion of influence for symmetric convex sets over Gaussian space, which we term "convex influence". We show that this new notion of influence shares many of the familiar properties of influences of variables for monotone Boolean functions $f: \{\pm1\}^n \to \{\pm1\}.$ Our main results for convex influences give Gaussian space analogues of many important results on influences for monotone Boolean functions. These include (robust) characterizations of extremal functions, the Poincar\'e inequality, the Kahn-Kalai-Linial theorem, a sharp threshold theorem of Kalai, a stability version of the Kruskal-Katona theorem due to O'Donnell and Wimmer, and some partial results towards a Gaussian space analogue of Friedgut's junta theorem. The proofs of our results for convex influences use very different techniques than the analogous proofs for Boolean influences over $\{\pm1\}^n$. Taken as a whole, our results extend the emerging analogy between symmetric convex sets in Gaussian space and monotone Boolean functions from $\{\pm1\}^n$ to $\{\pm1\}$

中文翻译：

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凸面影响

我们为高斯空间上的对称凸集引入了一个新的影响概念，我们称之为“凸影响”。我们表明，这种新的影响概念与单调布尔函数 $f 的变量影响的许多熟悉属性相同：\{\pm1\}^n \to \{\pm1\}.$ 我们对凸影响的主要结果给出对单调布尔函数影响的许多重要结果的高斯空间类似物。这些包括极值函数的（稳健的）表征、Poincar\'e 不等式、Kahn-Kalai-Linial 定理、Kalai 的尖锐阈值定理、由于 O'Donnell 和 Wimmer 的 Kruskal-Katona 定理的稳定性版本，以及弗里德古特军政府定理的高斯空间模拟的一些部分结果。我们对凸影响结果的证明使用了与 $\{\pm1\}^n$ 上的布尔影响的类似证明非常不同的技术。总的来说，我们的结果将高斯空间中对称凸集和单调布尔函数之间的新兴类比从 $\{\pm1\}^n$ 扩展到 $\{\pm1\}$