The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytope
Abstract
The problem of finding the largest number of points in the unit cross-polytope such that the l1-distance between any two distinct points is at least 2r is investigated for r∈(1-1n,1] in dimensions ≥2 and for r∈(12,1] in dimension 3. For the n-dimensional cross-polytope, 2n points can be placed when r∈(1-1n,1]. For the three-dimensional cross-polytope, 10 and 12 points can be placed if and only if r∈(35,23] and r∈(47,35] respectively, and no more than 14 points can be placed when r∈(12,47]. Also, constructive arrangements of points that attain the upper bounds of 2n, 10, and 12 are provided, as well as 13 points for dimension 3 when r∈(12,611].
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