Insecurity is generic in a conformal class of Riemannian metrics

A pair of points x, y   in a Riemannian manifold (M,g)(M,g) is said to be secure if there exists a finite set of points intercepting every geodesic segment joining x to y  . Given any conformal equivalence class CC of Riemannian metrics on a closed manifold M of dimension at least two and given any pair of points x, y in M  , there exists a dense GδGδ set C′⊂CC′⊂C such that x and y are not secure for every metric g   in C′C′.