Optimal Placement and Shape Design of Sensors via Geometric Criteria

The optimal placement and design of sensors is commonly encountered in industrial and applied problems, such as urban planning and the supervision of temperature and pressure in gas networks. In essence, sensors are considered optimally designed when they ensure the highest level of observation for the specific phenomenon in question. Typically, this design process is guided by specific objectives and is subject to constraints commonly defined by an appropriate partial differential equation (PDE), taking into account the underlying physics of the process. In the present work, we focus on two independent study cases: The optimal shape design of a convex sensor. • The optimal placement of a finite number of sensors inside a given region. Here, we address the problem in a purely geometric setting, without involving a specific PDE model. We consider a simple and natural geometric criterion of performance, based on distance functions. But, as we shall see, tackling it will require to employ geometric analysis methods.