Functions on the sphere with critical points in pairs and orthogonal geodesic chords

Using an estimate on the number of critical points for a Morse-even   function on the sphere SmSm, m≥1m≥1, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. This yields also a multiplicity result for brake orbits in a potential well.