A theoretical measure technique for determining $3\mathrm{D}$ symmetric nearly optimal shapes with a given center of mass

In this paper, a new approach is proposed for designing the nearly-optimal three dimensional symmetric shapes with desired physical center of mass. Herein, the main goal is to find such a shape whose image in $(r, \theta)$-plane is a divided region into a fixed and variable part. The nearly optimal shape is characterized in two stages. Firstly, for each given domain, the nearly optimal surface is determined by changing the problem into a measure-theoretical one, replacing this with an equivalent infinite dimensional linear programming problem and approximating schemes; then, a suitable function that offers the optimal value of the objective function for any admissible given domain is defined. In the second stage, by applying a standard optimization method, the global minimizer surface and its related domain will be obtained whose smoothness is considered by applying outlier detection and smooth fitting methods. Finally, numerical examples are presented and the results are compared to show the advantages of the proposed approach.