Abstract
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 $r\in\left(\frac{1}{2},\frac{6}{11}\right]$.
Highlights
Let K and L be origin-symmetric convex sets in Rn with nonempty interiors
In the k (K) n2 + n, and in the case when K is a cross-polytope the lower bound has been considerably improved to k (Cn∗)
9 (1−o(1))n 8 by [13] and to k (Cn∗) 1.13488(1−o(1))n by [19], which is the best known asymptotic lower bound for the cross-polytope
Summary
Let K and L be origin-symmetric convex sets in Rn with nonempty interiors. A set D ⊂ Rn is a (translative) packing set for K if, for all distinct x, y ∈ D,. In the k (K) n2 + n, and in the case when K is a cross-polytope the lower bound has been considerably improved to k (Cn∗). 9 (1−o(1))n 8 by [13] and to k (Cn∗) 1.13488(1−o(1))n by [19], which is the best known asymptotic lower bound for the cross-polytope. The upper bound for the number of points in the crosspolytope such that the distance between any two distinct points is at least 2r is linear in the dimension of the cross-polytope. Fejes Toth, Fodor, and Vıgh [5] describe some upper bounds for the packing density of Cn∗, including an asymptotic δ (Cn∗) C · 0.86850n for n 7 and a fixed constant C. Theorem 3 is proved in Section 5, and Section 6 presents a gallery of diagrams related to these lower bounds
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