In this talk we discuss the multiplicity question for prime closed orbits of a dynamically convex Reeb flow on the boundary of a $2n$-dimensional star-shaped domain. Our first main result asserts that such a flow has at least n prime closed Reeb orbits, settling a conjecture which is usually attributed to Ekeland. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly n or infinitely many prime closed orbits. This is a higher-dimensional contact variant of Franks' celebrated $2$-or-infinity theorem and, viewed from the symplectic dynamics perspective, settles a particular case of the contact Hofer-Zehnder conjecture. The talk is based on a joint work with Erman Cineli and Viktor Ginzburg.