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In this paper we prove that solutions to several shape optimization problemsin the plane, with a convexity constraint on the admissible domains, arepolygons. The main terms of the shape functionals we consider are either E f($\Omega$), the Dirichlet energy of the Laplacian in the domain $\Omega$, or$\lambda$ 1 ($\Omega$), the first eigenvalue of the Dirichlet-Laplacian.Usually, one considers minimization of such functionals (often with measureconstraint), as for example for the famous Saint-Venant and Faber-Krahninequalities. By adding the convexity constraint (and possibly otherconstraints to ensure existence of an optimal shape) one allows to consider therather unusual and difficult question of maximizing these functionals. Thispaper follows a series of papers by the authors, where the leading idea is thata certain concavity property of the shape functional that is minimized leadsoptimal shapes to locally saturate the convexity constraint, whichgeometrically means that they are polygonal. In these previous papers, theleading term in the shape functional usually was the opposite of the perimeter,for which the aforementioned concavity property was rather easy to obtainthrough computations of its second order shape derivative. By carryingclassical shape calculus, a similar concavity property can be observed for theopposite of E f or $\lambda$ 1 , for shapes that are smooth and convex. Themain novelty in the present paper is indeed the proof of a weak convexityproperty of E f and $\lambda$ 1 when it is only known that the shape is planarand convex, which means we have to consider rather non-smooth shapes. This workinvolves new computations and estimates of the second order shape derivative ofE f and $\lambda$ 1 that are interesting in themselves.

Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains