In this talk, I will explain the main components of the proof of the Log-Convex Density Conjecture. This conjecture, due to K. Brakke, asserts that balls centered at the origin are isoperimetric regions in Euclidean space endowed with a positive density which is smooth, radially symmetric, and log-convex. I will also show that these are the only isoperimetric regions, unless the density is constant on some ball.

These methods have recently been used to solve a similar problem; in Euclidean space with density f(x) = |x|^p, p > 0, balls whose boundaries pass through the origin are isoperimetric regions. I will explain how the components of the proof of the first theorem can be adapted to prove this one as well.