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问题。第一个问题是确定任意$\mathbb{R}^{d+1}$中的zonoid在Hausdorff距离下能被$n$个线段之和逼近的误差界。第二个问题是在其变分空间上为浅层ReLU$^k$神经网络确定均匀范数下的最优逼近速率。第一个问题在$d\neq 2,3$时已得到解决，但当$d=2,3$时，最佳上界与下界之间仍存在对数间隙。我们消除了这一间隙，从而在所有维度上完成了该问题的求解。对于第二个问题，我们的技术在$k\geq 1$时显著改进了现有逼近速率，并能够同时实现目标函数及其导数的均匀逼近。