  Under a convexity assumption on the boundary we solve a local inverse
problem, namely we show that the geodesic X-ray transform can be inverted
locally in a stable manner; one even has a reconstruction formula. We also show
that under an assumption on the existence of a global foliation by strictly
convex hypersurfaces the geodesic X-ray transform is globally injective. In
addition we prove stability estimates and propose a layer stripping type
algorithm for reconstruction.
