  The Hessian of the renormalized volume of geometrically finite hyperbolic
$3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$
with totally geodesic boundary of the convex core, is shown to be a strictly
positive bilinear form on the tangent space to Teichm\"uller space. The metric
$g$ is known from results of Bonahon and Storm to be an absolute minimum for
the volume of the convex core. We deduce the strict convexity of the functional
volume of the convex core at its minimum point.
