The first attempt to determine three dimensional Optimal shapes by means of Radon Measures

In this paper, a new measure theoretical approach is proposed to find a solution of optimization surface problems with a given image in two cases: presence and absence of obstacles. In addition, several basic states, such as being symmetric or asymmetric are also considered. Herein, the main goal is to find such a surface whose image in (r, θ)-plane is given. First, the problem is redefined in a variational form which can be expressed as an optimal control problem by introducing artificial controls. Then, for each given domain, the optimal surface is determined by changing the problem into a measure-theoretical one which replaces the problem with an equivalent infinite dimensional linear programming one. Finally, its related solution of a finite LP is used to construct a nearly optimal surface for the original problem in which its smoothness are considered by applying outlier detection and smooth fitting.