Disk-like surfaces of section and symplectic capacities

We prove that the cylindrical capacity of a dynamically convex domain in ${\mathbb{R}}^4$ agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in ${\mathbb{R}}^4$ which are sufficiently $C^3$ close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salomão establishing a systolic inequality for such domains.