Abstract. In this article, we consider the wave equation on a domain of IR n with Lipschitz boundary. For every observable subset Γ of the boundary ∂Ω (endowed with the usual Hausdorff measure H n−1 on ∂Ω), the observability constant provides an account for the quality of the reconstruction in some inverse problem. Our objective is here to determine what is, in some appropriate sense, the best observation domain. After having defined a randomized observability constant, more relevant tan the usual one in applications, we determine the optimal value of this constant over all possible subsets Γ of prescribed area H n−1 (Γ) = LH n−1 (∂Ω), with L ∈ (0, 1), under appropriate spectral assumptions on Ω. We compute the maximizers of a relaxed version of the problem, and then study the existence of an optimal set of particular domains Ω. We then define and study an approximation of the problem with a finite number of modes, showing existence and uniqueness of an optimal set, and provide some numerical simulations.1. Setting of the shape optimization problem 1.1. A brief state of the art.The literature on the optimal observation or sensor location problems is abundant in engineering applications, where mainly the optimal location of sensors or controllers is investigated. In many contributions, numerical tools are developed to solve a simplified version of the optimal design problem where either the partial differential equation has been replaced with a discrete approximation, or the class of optimal designs is replaced with a compact finite dimensional set (see for example [1,5,22] and [15] where such problems are investigated in a more general setting).From an engineering point of view, the aim is to optimize the number, the position or the shape of sensors in order to improve the estimation of the state of the system. Fields of applications are very numerous, for example structural acoustics, piezoelectric actuators, or vibration control in mechanical structures just to name a few of them, are interested in such a study.In this study, we model this issue by maximizing an observability constant with respect to the shape an location of the sensors. We refer to [21,23] for a general presentation of observability inequalities for wave like systems. We adopt here the "randomized" approach proposed in [18,19] Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx