High-dimensional envy-free partitions

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 $n$ observers 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 $n$ observers. We also prove the existence of convex partitions of ${\mathbb{R}}^d$ which allow for envy-free divisions among several groups of $n$ observers simultaneously.