*Accepted Paper*

**Inserted:** 12 may 2021

**Last Updated:** 30 aug 2021

**Journal:** Math. Eng.

**Pages:** 25

**Year:** 2021

**Abstract:**

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 $q=2$).

**Keywords:**
Torsional rigidity, Cheeger constant, Inradius, hidden convexity, convex duality, geometric estimates, Laplacian eigenvalues

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