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In this article, we consider the problems of finding in $d+1$ dimensions aminimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box anda minimum-volume convex body into which a given set of $d$-dimensionalunit-radius balls can be packed under translations. The computational problemis neither known to be NP-hard nor to be in NP. We give a constant-factorapproximation algorithm for each of these containers based on a reduction tofinding a shortest Hamiltonian path in a weighted graph, which in turn modelsthe problem of stabbing the centers of the input balls while keeping themdisjoint. 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 abyproduct, this implies that for $d \geq 2$ there is no finite size$(d+1)$-dimensional convex body into which all $d$-dimensional unit-radiusballs can be packed simultaneously.

Packing d-dimensional balls into a d+1-dimensional container