An elementary proof for the Double Bubble problem in ${\ell }^1$ norm

Abstract

We study the double bubble problem with perimeter taken with respect to the ${\ell }_1$ norm on ${\mathbb{R}}^2$. We give an elementary proof for the existence of minimizing sets for any volume ratio parameter $0<\alpha \le 1$ by direct comparison to a small family of parameterized sets. By simple analysis on this family we obtain the minimizing shapes found in Morgan et al 1998.

16 pages 14 figures