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The Hardy constant of a simply connected domain $\Omega\subset\R^2$ is thebest constant for the inequality \[ \int_{\Omega}|\nabla u|^2dx \geqc\int_{\Omega} \frac{u^2}{{\rm dist}(x,\partial\Omega)^2}\, dx \;, u\inC^{\infty}_c(\Omega). \] After the work of Ancona where the universal lowerbound 1/16 was obtained, there has been a substantial interest on computing orestimating the Hardy constant of planar domains. In this work we determine theHardy constant of an arbitrary quadrilateral in the plane. In particular weshow that the Hardy constant is the same as that of a certain infinitesectorial region which has been studied by E.B. Davies.

On the Hardy constant of non-convex planar domains: the case of the quadrilateral