For the l1 unit cross-polytope, the maximum number of points with pairwise l1-distance at least 2r is exactly 2n for r in (1-1/n, 1], and in three dimensions is exactly 10 for r in (3/5, 2/3] and 12 for r in (4/7, 3/5], with an upper bound of 14 for r in (1/2, 4/7].
On packing spheres into containers (about Kepler's finite sphere packing problem)
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abstract
In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can not have a ``simple structure'' for large $n$. By this we in particular find that there exist arbitrary small $r>0$, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius $r$, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing ``will be the tightest possible, so that in no other arrangement more spheres could be packed into the same container''.
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math.MG 1years
2019 1verdicts
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The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope
For the l1 unit cross-polytope, the maximum number of points with pairwise l1-distance at least 2r is exactly 2n for r in (1-1/n, 1], and in three dimensions is exactly 10 for r in (3/5, 2/3] and 12 for r in (4/7, 3/5], with an upper bound of 14 for r in (1/2, 4/7].