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

Parker Duncan; Rory O'Dwyer; Eviatar B. Procaccia

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arXiv · 2008.07767

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

Parker Duncan·Rory O'Dwyer·Eviatar B. Procaccia

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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.

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Parker Duncan, Rory O'Dwyer, Eviatar B. Procaccia. 2020-08-18. An elementary proof for the Double Bubble problem in $\ell^1$ norm. https://arxiv.org/abs/2008.07767

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