# Double bubbles minimize

### Abstract

The classical isoperimetric inequality in $\mathbb{R}^3$ states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of $120^{\circ}$.

DOI

https://doi.org/10.2307/121042

Joel Hass

Roger Schlafly