Optimal Design of Sensors via Geometric Criteria

Abstract

We consider a convex set $$\Omega $$ Ω and look for the optimal convex sensor $$\omega \subset \Omega $$ ω ⊂ Ω of a given measure that minimizes the maximal distance to the points of $$\Omega .$$ Ω . This problem can be written as follows $$\begin{aligned} \inf \{d^H(\omega ,\Omega ) \ |\ |\omega |=c\ \text {and}\ \omega \subset \Omega \}, \end{aligned}$$ inf { d H ( ω , Ω ) | | ω | = c and ω ⊂ Ω } , where $$c\in (0,|\Omega |),$$ c ∈ ( 0 , | Ω | ) , $$d^H$$ 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.