The Double Bubble Problem in the Hexagonal Norm

We study the double bubble problem over connected sets where the perimeter is taken with respect to the hexagonal norm, i.e. the norm whose unit circle in R2 is the regular hexagon. We provide an elementary proof for the existence of minimizing sets for volume ratio parameter α∈(0,1] by arguing that any minimizer must belong to a small family of parameterized sets. This family is further simplified by showing that 60∘ angles are not optimal as well as other geometric exclusions. We then provide a minimizer for all α∈(0,1] except at a single point, for which we find two minimizing configurations.