Dynamical convexity and elliptic periodic orbits for Reeb flows

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A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in R²ⁿ 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 S³. Applications include geodesic flows under pinching conditions, magnetic flows and toric contact manifolds.