  We study the boundary rigidity problem with partial data consisting of
determining locally the Riemannian metric of a Riemannian manifold with
boundary from the distance function measured at pairs of points near a fixed
point on the boundary. We show that one can recover uniquely and in a stable
way a conformal factor near a strictly convex point where we have the
information. In particular, this implies that we can determine locally the
isotropic sound speed of a medium by measuring the travel times of waves
joining points close to a convex point on the boundary.
  The local results lead to a global lens rigidity uniqueness and stability
result assuming that the manifold is foliated by strictly convex hypersurfaces.
