Isoperimetric 3- and 4-bubble results on $\mathbb{R}$ with density $|x|$

Abstract

We study the isoperimetric problem on ${\mathbb{R}}^1$ with a prescribed density function $f(x)=|x|$. Under these conditions, we find that isoperimetric $3$-bubble and $4$-bubble results satisfy a regular structure. As our regions increase in size, the intervals that form them alternate back-and-forth across the origin, with the smaller regions closer to the origin. This expands on previously known observations about the single- and double-bubble results on $\mathbb{R}$ with density $|x{|}^p$.

1 figure. arXiv admin note: text overlap with arXiv:2201.01808