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Manfred Weis
Manfred Weis
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define a box packing as gap-less if

  • all inner boxes have disjoint interior
  • the sum of volumes of the iinerinner 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.

define a box packing as gap-less if

  • all inner boxes have disjoint interior
  • the sum of volumes of the iiner 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.

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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Manfred Weis
Manfred Weis
  • 13.2k
  • 4
  • 34
  • 76

Stable gap-less packing of a box with boxes

define a box packing as gap-less if

  • all inner boxes have disjoint interior
  • the sum of volumes of the iiner 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.