Different cuboids into a cuboid Determine the minimum number of
cuboid containers (bins) that are required to pack a given set of item cuboids. The rectangular cuboids to be packed can be rotated by 90 degrees on each axis.
Spheres into a Euclidean ball The problem of finding the smallest ball such that
disjoint open
unit balls may be packed inside it has a simple and complete answer in -dimensional Euclidean space if k \leq n+1, and in an infinite-dimensional
Hilbert space with no restrictions. It is worth describing in detail here, to give a flavor of the general problem. In this case, a configuration of pairwise
tangent unit balls is available. People place the centers at the vertices a_1, \dots, a_k of a regular (k-1) dimensional
simplex with edge 2; this is easily realized starting from an
orthonormal basis. A small computation shows that the distance of each vertex from the barycenter is \sqrt{2\big(1-\frac{1}{k} \big)}. Moreover, any other point of the space necessarily has a larger distance from
at least one of the vertices. In terms of inclusions of balls, the open unit balls centered at a_1, \dots, a_k are included in a ball of radius r_k := 1+\sqrt{2\big(1-\frac{1}{k}\big)}, which is minimal for this configuration. To show that this configuration is optimal, let x_1, \dots, x_k be the centers of disjoint open unit balls contained in a ball of radius centered at a point x_0. Consider the
map from the finite set \{x_1,\dots,x_k\} into \{a_1,\dots,a_k\} taking x_j in the corresponding a_j for each 1 \leq j \leq k. Since for all 1 \leq i , \|a_i-a_j\| = 2\leq\|x_i-x_j\| this map is 1-
Lipschitz and by the
Kirszbraun theorem it extends to a 1-Lipschitz map that is globally defined; in particular, there exists a point a_0 such that for all 1\leq j\leq k one has \|a_0-a_j\| \leq \|x_0-x_j\|, so that also r_k\leq 1+\|a_0-a_j\|\leq 1+\|x_0-x_j\| \leq r. This shows that there are disjoint unit open balls in a ball of radius
if and only if r \geq r_k. Notice that in an infinite-dimensional Hilbert space this implies that there are infinitely many disjoint open unit balls inside a ball of radius if and only if r\geq 1+\sqrt{2}. For instance, the unit balls centered at \sqrt{2}e_j, where \{e_j\}_j is an orthonormal basis, are disjoint and included in a ball of radius 1 + \sqrt{2} centered at the origin. Moreover, for r , the maximum number of disjoint open unit balls inside a ball of radius is \left\lfloor \frac{2}{2-(r-1)^2}\right\rfloor.
Spheres in a cuboid People determine the number of spherical objects of given diameter that can be packed into a cuboid of size a \times b \times c.
Identical spheres in a cylinder People determine the minimum height of a
cylinder with given radius that will pack identical spheres of radius . For a small radius the spheres arrange to ordered structures, called
columnar structures.
Polyhedra in spheres People determine the minimum radius that will pack identical, unit
volume polyhedra of a given shape. ==Packing in 2-dimensional containers==