On closed characteristics of minimal action on a convex three-sphere

We prove that every closed characteristic of minimal action on the boundary of a uniformly convex domain in ${\mathbb{R}}^4$ bounds a disk-like global surface of section. A corollary is that the cylindrical symplectic capacity of a convex body in ${\mathbb{R}}^4$ coincides with the minimal action of a closed generalized characteristic on its boundary.