Estimates for the energy of the solutions to elliptic Dirichlet problems on convex domains

We provide an estimate of the energy of the solutions to the Poisson equation with constant data and Dirichlet boundary conditions in a convex domain $\Omega\subset\R^n$. This estimate is obtained by restricting the variational formulation of the problem to the space of functions depending only on the distance from the boundary of $\Omega$. The main tool in the proof is an isoperimetric inequality for convex domains, which is a consequence of the Brunn-Minkowski theorem.