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.

翻译：我们研究以下两个相关的问题。第一个问题是确定在$\mathbb{R}^{d+1}$中任意一个zonoid如何通过$n$条线段的和在Hausdorff距离下被逼近到何种误差。第二个问题是确定浅层ReLU$^k$神经网络在其变分空间上的均匀范数下的最优逼近率。第一个问题在$d\neq 2,3$时已得到解决，但当$d=2,3$时，最佳上下界之间存在一个对数间隙。我们填补了这一间隙，从而完成了所有维数下的解答。对于第二个问题，当$k\geq 1$时，我们的技术显著改进了现有的逼近率，并能够实现对目标函数及其导数的均匀逼近。