Isoperimetric 3- and 4-bubble results on $\mathbb{R}$ with density $|x|$

Evan Alexander; Emily Burns; John Ross; Jesse Stovall; Zariah Whyte

Isoperimetric 3- and 4-bubble results on $\mathbb{R}$ with density $|x|$

Metric Geometry 2022-01-10 v1

Authors: Evan Alexander , Emily Burns , John Ross , Jesse Stovall , Zariah Whyte

Abstract

We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f(x) = |x|$ . Under these conditions, we find that isoperimetric $3$ -bubble and $4$ -bubble results satisfy a regular structure. As our regions increase in size, the intervals that form them alternate back-and-forth across the origin, with the smaller regions closer to the origin. This expands on previously known observations about the single- and double-bubble results on $\mathbb{R}$ with density $|x|^p$ .

Keywords

additive number theory regularity theory sphere packings

Cite

@article{arxiv.2201.02197,
  title  = {Isoperimetric 3- and 4-bubble results on $\mathbb{R}$ with density $|x|$},
  author = {Evan Alexander and Emily Burns and John Ross and Jesse Stovall and Zariah Whyte},
  journal= {arXiv preprint arXiv:2201.02197},
  year   = {2022}
}

Comments

1 figure. arXiv admin note: text overlap with arXiv:2201.01808

Isoperimetric 3- and 4-bubble results on $\mathbb{R}$ with density $|x|$

Abstract

Keywords

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