  We study the following two related problems. The first is to determine to
what error an arbitrary zonoid in $\mathbb{R}^{d+1}$ can be approximated in the
Hausdorff distance by a sum of $n$ line segments. The second is to determine
optimal approximation rates in the uniform norm for shallow ReLU$^k$ neural
networks on their variation spaces. The first of these problems has been solved
for $d\neq 2,3$, but when $d=2,3$ a logarithmic gap between the best upper and
lower bounds remains. We close this gap, which completes the solution in all
dimensions. For the second problem, our techniques significantly improve upon
existing approximation rates when $k\geq 1$, and enable uniform approximation
of both the target function and its derivatives.
