  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-Salom\~{a}o establishing a systolic
inequality for such domains.
