On local packings of the cross-polytope (Q2194090)

Summary: The problem of finding the largest number of points in the unit cross-polytope such that the \(l_1\)-distance between any two distinct points is at least \(2r\) is related to packings. For the \(n\)-dimensional cross-polytope, we show that \(2n\) points can be placed when \(r\in\left(1-\frac{1}{n},1\right]\). For the three-dimensional cross-polytope, \(10\) and \(12\) points can be placed if and only if \(r\in\left(\frac{3}{5},\frac{2}{3}\right]\) and \(r\in\left(\frac{4}{7},\frac{3}{5}\right]\) respectively, and no more than \(14\) points can be placed when \(r\in\left(\frac{1}{2},\frac{4}{7}\right]\). 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 \(r\in\left(\frac{1}{2},\frac{6}{11}\right]\).