Author’s abstract: The Hessian of the renormalized volume of geometrically finite hyperbolic 3-manifolds without rank-1 cusps, computed at the hyperbolic meric \(g_{\mathrm{geod}}\) with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric \(g_{\mathrm{geod}}\) 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 functional volume of the convex core at the minimum point.