Isoperimetric 3- and 4-Bubble Results on R With Density |x|

Abstract

We study the isoperimetric problem on ℝ¹ 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 ℝ with density |x|^p.