The Log-Convex Density Conjecture for radial densities

Let $\mathbb{R}^{n}$, with $n\geq2$, carry a radial simple density, meaning that the surface and volume densities agree and depend only on the distance from the origin. A density is log-convex when its radial density function has a convex logarithm.

Log-Convex Density Conjecture. In $\mathbb{R}^{n}$ with radial log-convex simple density, balls about the origin are isoperimetric for every volume.

This conjecture concerns the isoperimetric problem in Euclidean space with radial density: among regions enclosing a prescribed weighted volume, balls centered at the origin should minimize weighted perimeter. The source notes that log-convexity is equivalent to stability of these balls, while additional regularity is needed to exclude singular or irregular examples; the general conjecture remains unclear.