An Elementary Proof for the Double Bubble Problem in $$\ell ^1$$ Norm

We study the double bubble problem with perimeter taken respect to $$\ell _1$$ norm on $${{\mathbb {R}}}^2$$ . give an elementary proof for existence of minimizing sets any volume ratio parameter $$0<\alpha \le 1$$ by direct comparison a small family parameterized sets. By simple analysis this family, we obtain shapes found in Morgan et al. J Geom Anal 8(1):97-*115 , 1998.