*Accepted Paper*

**Inserted:** 2 feb 2017

**Last Updated:** 20 nov 2017

**Journal:** ESAIM: COCV

**Year:** 2017

**Doi:** https://doi.org/10.1051/cocv/2017069

**Abstract:**

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider $T(\Omega)/(M(\Omega)

\Omega

)$ and $M(\Omega)\lambda_1(\Omega)$, where $\Omega$ is a bounded open set of $\mathbb R^d$ with finite Lebesgue measure $

\Omega

$, $M(\Omega)$ denotes the maximum of the torsion function, $T(\Omega)$ the torsion, and $\lambda_1(\Omega)$ the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

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