Plane Segmentation for Proportional Division of Resources

Fair allocation is a significant research area in mathematics, economics, and artificial intelligence, involving the proportional allocation of heterogeneous goods, such as land, among several agents with varying preferences. Proportionality is one of the common principles of fairness. This paper focuses on the proportional division of the plane. We demonstrate the existence of a solution that meets proportionality by utilizing complementary slackness techniques, leading to optimal regions bounded by a specific curve. Our approach simplifies the problem to convex optimization and provides a subgradient descent algorithm for proportional plane segmentation.