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

TL;DR

This paper provides an elementary proof for the existence and characterization of minimal double bubble shapes in the plane under the $\, ext{l}_1$ norm, confirming known shapes through direct comparison methods.

Contribution

It offers a new, elementary proof for the double bubble problem in $\, ext{l}_1$ norm, simplifying previous approaches and explicitly identifying minimizing shapes.

Findings

Existence of minimizing sets for all volume ratios $0<\,\alpha\le1$

Explicit characterization of minimizing shapes

Simplified proof method compared to prior work

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 $0<\alpha\le1$ 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.