  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.
