Pareto Optimality and Isoperimetry

TL;DR

This paper explores a new class of geometric extremal problems involving conflicting goals, such as maximizing volume while minimizing width, using multi-criteria decision making approaches.

Contribution

It introduces a novel framework for solving geometrical extremal problems with multiple conflicting objectives based on Pareto optimality and isoperimetric principles.

Findings

Identification of Pareto optimal solutions in geometric problems

Development of methods for balancing conflicting geometric criteria

Application of multi-criteria decision making to convex geometry

Abstract

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.