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

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 ${\left[\log f\right]}^{\prime \prime }\le 0$. 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.