arXiv subjects

Parker Duncan

Publications and source records attributed to Parker Duncan.

3 recordsLinked to original sources

arXiv2024

The Double Bubble Problem in the Hexagonal Norm

We study the double bubble problem where the perimeter is taken with respect to the hexagonal norm, i.e. the norm whose unit circle in $\mathbb{R}^2$ is the regular hexagon. We provide an elementary proof for the existence of minimizing sets for volume ratio parameter $α\in (0,1]$ by arguing that any minimizer must belong to a small family of parameterized sets. This family is further simplified by showing that $60^{\circ}$ angles are not optimal as well as other geometric exclusions. We then provide a minimizer for all $α\in(0,1]$ except at a single point, for which we find two minimizing configurations.

math.MG↗

arXiv2021

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

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}.

math.MG↗

arXiv2020

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

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 $0<α\le1$ 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.

math.GT↗