In this article, we consider the problems of finding in $d+1$ dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of $d$-dimensional unit-radius balls can be packed under translations. The computational problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm for each of these containers based on a reduction to finding a shortest Hamiltonian path in a weighted graph, which in turn models the problem of stabbing the centers of the input balls while keeping them disjoint. We also show that for $n$ such balls, a container of volume $O(n^{\frac{d-1}{d}})$ is always sufficient and sometimes necessary. As a byproduct, this implies that for $d \geq 2$ there is no finite size $(d+1)$-dimensional convex body into which all $d$-dimensional unit-radius balls can be packed simultaneously.

翻译：本文研究在d+1维空间中寻找最小体积的轴对齐盒、最小体积的任意方向盒以及最小体积的凸体，使得给定的d维单位半径球集能在平移下装入其中。该计算问题既未被证明是NP难的，也未被证明属于NP。我们通过将问题归约到加权图中最短哈密顿路径的求解（该路径模型模拟了在保持输入球不相交的同时刺穿其中心的问题），为每种容器给出了常数因子近似算法。我们还证明了对于n个这样的球，体积为$O(n^{\frac{d-1}{d}})$的容器总是充分的，且有时是必要的。作为副产品，这一结果表明对于$d \geq 2$，不存在有限大小的(d+1)维凸体能同时装入所有d维单位半径球。