Approximate Gaussian isoperimetry for k sets

TL;DR

This paper investigates the minimal Gaussian boundary measure for partitions of Euclidean space and spheres into k equal parts, revealing it scales with the square root of the logarithm of k.

Contribution

It establishes a new approximate isoperimetric inequality for k-partitions in Gaussian space and on the sphere, extending classical results to multiple sets.

Findings

Boundary measure scales as √(log k) for Gaussian partitions.

Similar results hold for partitions of the sphere.

Provides bounds for minimal boundary measures in high-dimensional spaces.

Abstract

Given $2\le k\le n$, the minimal $(n-1)$-dimensional Gaussian measure of the union of the boundaries of $k$ disjoint sets of equal Gaussian measure in $\R^n$ whose union is $\R^n$ is of order $\sqrt{\log k}$. A similar results holds also for partitions of the sphere $S^{n-1}$ into $k$ sets of equal Haar measure.