  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.
