Combinatorial Theory

A vast array of envy-free results have been found for the subdivision of one-dimensional resources, such as the interval \([0,1]\). The goal is to divide the space into \(n\) pieces and distribute them among \(n\) guests such that each receives their favorite pieces. We study high-dimensional versions of these results. We prove that several spaces of convex partitions of \(\mathbb{R}^d\) allow for envy-free division among any \(n\) guests. We also prove the existence of convex partitions of \(\mathbb{R}^d\) which allow envy-free divisions among several groups of \(n\) guests simultaneously.

Mathematics Subject Classifications: 91B32, 52A37, 55M20, 28A75

Keywords: Mass partition, KKM cover, Envy-free partition, Equivariant topology, Voronoi diagram