  A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow
of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed
orbit. Two important contributions toward its proof were given by Ekeland in
1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex
hypersurfaces satisfying suitable pinching conditions and for antipodal
invariant convex hypersurfaces respectively. In this work we present a
generalization of these results using contact homology and a notion of
dynamical convexity first introduced by Hofer-Wysocki-Zehnder for tight contact
forms on $S^3$. Applications include geodesic flows under pinching conditions,
magnetic flows and toric contact manifolds.
