# Eldan's stochastic localization and tubular neighborhoods of complex-analytic sets

## TL;DR

This paper establishes a lower bound on the Gaussian measure of tubular neighborhoods around complex-analytic sets, relating them to affine subspaces, using Eldan's stochastic localization technique.

## Contribution

It introduces a novel application of Eldan's stochastic localization to compare Gaussian measures of neighborhoods of complex-analytic sets and affine subspaces.

## Key findings

- Gaussian measure of neighborhoods of Z is at least that of E
- The result applies to zero sets of holomorphic maps from C^n to C^k
- Provides a new geometric measure comparison for complex-analytic sets

## Abstract

Let Z be the zero set of a holomorphic map from C^n to C^k. Assume that Z is non-empty. We prove that for any r > 0, the Gaussian measure of the Euclidean r-neighborhood of Z is at least as large as the Gaussian measure of the Euclidean r-neighborhood of E, where E is any (n-k)-dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.