The infimum of the dual volume of convex cocompact hyperbolic 3-manifolds

We show that the infimum of the dual volume of the convex core of a convex co-compact hyperbolic 3-manifold with incompressible boundary coincides with the infimum of the Riemannian volume of its convex core, as we vary the geometry by quasi-isometric deformations. We deduce a linear lower bound of the volume of the convex core of a quasi-Fuchsian manifold in terms of the length of its bending measured lamination, with optimal multiplicative constant.