A Two-Sided Estimate for the Gaussian Noise Stability Deficit


October 20, 2013


The Gaussian Noise Stability of a set A in Euclidean space is the probability that for a Gaussian vector X conditioned to be in A, a small Gaussian perturbation of X will also be in A. Borel’s celebrated Isoperimetric inequality states that a half-space maximizes noise stability among sets with the same Gaussian measure. We will present a novel short proof of this inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: We show that the deficit between the noise stability of a set A and an equally probable half-space H can be controlled by a function of the distance between the corresponding centroids. As a consequence, we prove a conjecture by Mossel and Neeman, who used the total-variation distance.


Ronen Eldan

Ronen Eldan Received is B.A degree in Maths from the Open University of Israel in 2005, which was later extended to an additional discipline in Physics in 2006, at the Tel Aviv University. In 2012, He graduated his Ph.D. studies in mathematics at the Tel Aviv University, under the supervision of Prof. V. Milman and Prof. B. Klartag, specializing in probability, high dimensional convex geometry and computational geometry. Before and during most of his studies, he also worked as a computer security specialist, applied mathematician and team-leader of a programming group for the IDF and in the industry. He was an intern in the Theory Group at Microsoft research in Autumn 2011. Currently, he is a Post-Doctoral fellow at the department of Mathematics and Computer Science in the Weizmann Institute of Science.