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define a box packing as gap-less if

  • all inner boxes have disjoint interior
  • the sum of volumes of the inner box equals that of the outer box
  • the sum of the extents of the inner boxes in each principal direction equals that of the outer box

define a box paxking as stable if

  • for every hyperplane with a point inside the outer box there is an inner box with inner points to both sides of that hyperplane.

Questions:

  • for which $k$, depending on dimension $n$ of the boxes do stable gap-less packings of boxes with boxes exist?
  • how can such packings be found, given $k$ and $n$?

In $2{-}d$ there are many examples from squaring the square which is a very special case of this question for $n=2$; this question allows boxes of equal size and arbitrary extents in each principal direction.

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  • $\begingroup$ 1) What is $k$ in thefirst question? 2) What is a box —-a parallelepiped or a cube? 3) What is extent? $\endgroup$
    – Ilya Bogdanov
    Commented Oct 15 at 22:42
  • $\begingroup$ $k$ is the number of smaller boxes that are to be packed in a single bigger box; a box is the Cartesian product of $n$ intervals, one for each coordinate. Extent is the set of interval-lengths; in 3D aka length,width and height. $\endgroup$
    – Manfred Weis
    Commented Oct 16 at 2:45
  • $\begingroup$ But then, the sum of extents of the inner boxes in one direction should be larger than the extent of the outer box, except for the trivial cases… $\endgroup$
    – Ilya Bogdanov
    Commented Oct 16 at 6:39
  • $\begingroup$ @IlyaBogdanov what does "but them" refer to? Have a look at the suaring the square.illustration on the wikipedia page; the sum of inner extents equals the outer extents. We are consudering mathematical abstractions of boxes, not physical objects $\endgroup$
    – Manfred Weis
    Commented Oct 16 at 9:37

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