Pareto Optimality and Isoperimetry

Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body $\mathfrak{x}$, we try to maximize the volume of $\mathfrak{x}$ and minimize the width of $\mathfrak{x}$ simultaneously. These problems are addressed along the lines of multiple criteria decision making.