Double bubble theorem

In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble: three spherical surfaces meeting at angles of 120° on a common circle. The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989, but was not proven until 2002. The proof combines multiple ingredients. Compactness of rectifiable currents (a generalized definition of surfaces) shows that a solution exists. A symmetry argument proves that the solution must be a surface of revolution, and it can be further restricted to having a bounded number of smooth pieces. Jean Taylor proof of Plateau's laws describes how these pieces must be shaped and connected to each other, and a final case analysis shows that, among surfaces of revolution connected in this way, only the standard double bubble has locally-minimal area. The double bubble theorem extends the isoperimetric inequality, according to which the minimum-perimeter enclosure of any area is a circle, and the minimum-surface-area enclosure of any single volume is a sphere. Analogous results on the optimal enclosure of two volumes generalize to weighted forms of surface energy, to Gaussian measure of surfaces, and to Euclidean spaces of any dimension. According to the isoperimetric inequality, the minimum-perimeter enclosure of any area is a circle, and the minimum-surface-area enclosure of any single volume is a sphere. The existence of a shape with bounded surface area that encloses two volumes is obvious: just enclose them with two separate spheres. It is less obvious that there must exist some shape that encloses two volumes and has the minimum possible surface area: it might instead be the case that a sequence of shapes converges to a minimum (or to zero) without reaching it.