On-line packing sequences of segments, cubes and boxes (Q1368016)

The action takes place in \(d\)-dimensional Euclidean space. A packing of a sequence of convex bodies into a set \(C\) places them (into \(C\), each by an isometry) to have mutually non-intersecting interiors. The packing is translative if all the isometries are translations and it is one-line if each body is played in turn never to be replaced. Theorem: Let \(D\) be a box (rectangular parallelepiped) with integer sides and let \(q\geq 2\) be an integer. Any sequence of boxes with edge lengths from the set \(\{q^0, q^{-1}, q^{-2}, \dots\}\) whose total volume is at most \(\text{vol} (D)- q\{({q \over q-1})^{d-1} -1\}\) permits a translative on-line packing into \(D\); if all the boxes are cubes the packing is possible when the total volume is at most \(\text{vol} (D)\). In this paper this is one of 4 lemmas and 4 theorems which, taken together, add substantially to the subject.