Optimal Design of Sensors via Geometric Criteria

We consider a convex set Ω and look for the optimal convex sensor ω⊂ Ω of a given measure that minimizes the maximal distance to the points of Ω. This problem can be written as follows inf{dH(ω,Ω)||ω|=candω⊂Ω}, where c∈ (0 , | Ω |) , d^H being the Hausdorff distance. We show that the parametrization via the support functions allows us to formulate the geometric optimal shape design problem as an analytic one. By proving a judicious equivalence result, the shape optimization problem is approximated by a simpler minimization problem of a quadratic function under linear constraints. We then present some numerical results and qualitative properties of the optimal sensors and exhibit an unexpected symmetry breaking phenomenon.