Discrete $\ell^{1}$ Double Bubble solution is at most ceiling +2 of the continuous solution

TL;DR

This paper establishes that the optimal discrete double bubble solution on the integer grid is within two units above the continuous solution, providing bounds for the discrete case based on the continuous problem.

Contribution

It proves a tight bound relating discrete and continuous double bubble solutions in the $ ext{l}^1$ norm, extending understanding of geometric optimization on integer lattices.

Findings

Discrete solution is at most ceiling + 2 of the continuous solution

Provides bounds for discrete double bubble problem on $ ext{Z}^2$

Connects discrete and continuous geometric optimization problems

Abstract

In this paper we show that the solution of the discrete Double Bubble problem over $\mathbb{Z}^2$ is at most the ceiling function plus two of the continuous solution to the Double Bubble problem, with respect to the $\ell^1$ norm, found in \cite{morgan1998wulff} and \cite{duncan2020elementary}.