Eldan’s Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets

Abstract

Let $f: \mathbb {C}^n \rightarrow \mathbb {C}^k$ be a holomorphic function and set $Z = f^{-1}(0)$. Assume that Z is non-empty. We prove that for any $r > 0$,

$$\begin{aligned} \gamma _n(Z + r) \ge \gamma _n(E + r), \end{aligned}$$

where $Z + r$ is the Euclidean r-neighborhood of Z; $\gamma _n$ is the standard Gaussian measure in $\mathbb {C}^n$, and $E \subseteq \mathbb {C}^n$ is an $(n-k)$-dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.