Geometric influences
Geometric influences
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几何影响
作者：
Nathan Keller;Elchanan Mossel;Arnab Sen
我们在连续分布的产品空间中提出了一个新的定义，我们的定义是几何的，对于单调集合，它与统一扩展的边界的测量相同。 ）和塔格兰（Talagrand）的影响力对新定义的界限。尤其是坐标。我们建立了KKL结合的以下紧密类似：对于高斯测量t中的任何集合，存在坐标I √logn/n，其中c是通用常数。
We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogues of the Kahn-Kalai-Linial (KKL) and Talagrand’s influence sum bounds for the new definition. We further prove an analogue of a result of Friedgut showing that sets with small “influence sum” are essentially determined by a small number of coordinates. In particular, we establish the following tight analogue of the KKL bound: for any set in Rn of Gaussian measure t, there exists a coordinate i such that the i-th geometric influence of the set is at least ct(1−t) √ logn/n, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on Rn and the class of sets invariant under transitive permutation group of the coordinates.