# Gaussian bubble clusters

Joe Neeman June 15, 2020Suppose I ask you to divide \(\mathbb{R}^n\) into two pieces of fixed Gaussian
measure so that the surface area of the boundary is as small as possible. The
*Gaussian isoperimetric inequality* states that the best way to do it is by
cutting \(\mathbb{R}^n\) with a hyperplane:

Now what if I ask for *three* parts instead of two? That is: divide
\(\mathbb{R}^n\) into three pieces of fixed Gaussian measure so as to minimize
the surface area of the boundary. I solved this
recently with Emanuel Milman; the answer is
what we call a "tripod" partition (a.k.a. the "standard Y" or the "peace sign"
partition):

Ok, but if there are three parts, then why is this called a Gaussian
*double*-bubble? It's from analogy with the (more famous) *Euclidean*
double-bubble problem, which asks for a surface-area-minimizing
partition of \(\mathbb{R}^n\)
into three pieces with given Lebesgue measure (one of the measures is
necessarily infinite). We're definitely justified in calling this
a double-bubble problem, because the answer looks like one:

In the words of all those annoying standardized tests: the Gaussian double bubble is to the Gaussian isoperimetric inequality as the Euclidean double bubble is to the Euclidean isoperimetric inequality. That's why we call it a Gaussian double bubble.

What about more bubbles? In general, multi-bubble problems seem to be hard: the solution to the Euclidean triple-bubble problem is known only in two dimensions, and the Euclidean quadruple-bubble is not understood in any setting. However, a follow-up paper by Emanuel Milman and I solves the Gaussian \(k\)-bubble problem in \(n\) dimensions (i.e. we can find the optimal split of \(\mathbb{R}^n\) into \(k+1\) parts) whenever \(n \ge k\). For example, here's an optimal Gaussian triple bubble in three dimensions: