  We consider the sharp Sobolev-Poincar\'e constant for the embedding of
$W^{1,2}_0(\Omega)$ into $L^q(\Omega)$. We show that such a constant exhibits
an unexpected dual variational formulation, in the range $1<q<2$. Namely, this
can be written as a convex minimization problem, under a divergence--type
constraint. This is particularly useful in order to prove lower bounds. The
result generalizes what happens for the torsional rigidity (corresponding to
$q=1$) and extends up to the case of the first eigenvalue of the
Dirichlet-Laplacian (i.e. to $q=2$).
