  A fundamental object in a hyperbolic 3-manifold M is its convex core C(M),
defined as the smallest closed non-empty convex subset of M. We investigate the
way the geometry of the boundary S of C(M) varies as we vary the hyperbolic
metric of M. Thurston observed that the intrinsic metric of S is hyperbolic,
and that its bending is described by a measured geodesic lamination. We show
that, as the hyperbolic metric of the 3--manifold M varies differentiably, the
hyperbolic metric of the surface S varies in a C^1, but usually not C^2,
manner. Differentiability properties for the bending measured lamination are
conceptually less simple, because the space ML(S) of measured geodesic
laminations on S has no natural differentiable structure. However, ML(S) is a
piecewise linear manifold, and we also show that the bending measured geodesic
lamination varies differentiably in a piecewise linear sense.
  The two results are proved simultaneously, mixing the differentiable and
piecewise linear contexts. In particular, the 'corners' of the piecewise linear
structure of ML(S) account for the fact that the metric of S is not a C^2
function of the metric of M. A by-product of the proof is a similar
differentiablity property for Thurston's parametrization of complex projective
structures on a surface by hyperbolic metrics and measured laminations.
