Solution to the isoperimetric $n$-bubble problem on $\mathbb{R}^1$ with log-concave density

TL;DR

This paper characterizes the structure of isoperimetric n-bubbles on the real line with symmetric, log-concave densities that vanish at the origin, generalizing previous results for specific density functions.

Contribution

It provides a comprehensive analysis of the isoperimetric problem for a broad class of log-concave densities on , identifying the structure of optimal partitions for any number of bubbles.

Findings

Isoperimetric n-bubbles have a regular structure under the specified conditions.

The results generalize previous work on  with ^p density.

Contrasts with the behavior under log-convex densities, which lack such structure.

Abstract

We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f$ that affects how area and perimeter are measured. We examine density functions that are symmetric, radially increasing, and satisfy two additional conditions: they have a point of zero density (at the origin), and they satisfy a "log-concavity" requirement $\left[ \log f \right]'' \leq 0$. Under these conditions, we find that isoperimetric $n$-bubbles satisfy a regular structure and can be identified for arbitrary $n$. This generalizes recent work done on the density function $|x|^p$, and stands in contrast to log-convex density functions which have no such regular structure.