On local packings of the cross-polytope

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− 1 n , 1 ] in dimensions n > 2 and for r ∈ ( 1 2 , 1 ] in dimension 3. For the n-dimensional cross-polytope, 2n points can be placed when r ∈ ( 1− 1 n , 1 ] . For the three-dimensional cross-polytope, 10 and 12 points can be placed if and only if r ∈ ( 3 5 , 2 3 ] and r ∈ ( 4 7 , 3 5 ] respectively, and no more than 14 points can be placed when r ∈ ( 1 2 , 4 7 ] . 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 ∈ ( 1 2 , 6 11 ] . Mathematics Subject Classifications: 05B40, 52C17


Introduction
Let K and L be origin-symmetric convex sets in R n with nonempty interiors. A set D ⊂ R n is a (translative) packing set for K if, for all distinct x, y ∈ D, (x + int (K)) ∩ (y + int (K)) = ∅, where int (K) is the interior of K. For r > 0, we consider the problem of finding the maximum number of points in a packing set D of rK that is contained in L. This quantity will be denoted by γ (L, K, r) := max {|D| | D ⊂ L and ||x − y|| K 2r for any x, y ∈ D, x = y} , where ||x|| K = min {λ | λ 0 and x ∈ λK} and for a set S, its cardinality is denoted by |S|. If K = L then we use the notation γ (K, r) as a shorthand for γ (L, K, r). We will only deal with the situation where both K and L are the unit cross-polytope C * n = {x ∈ R n | n i=1 |x i | 1 }. A set D in C * n with k points such that the l 1 -distance between any two distinct points is greater than or equal to 2r is equivalent to a packing set for rC * n such that each set x = rC * n , x ∈ D, is contained inside the set (1 + r) C * n . Unless otherwise specified, we will use "distance" to mean the l 1 -distance. The vertices of C * n are the 2n unit vectors {±e i | i ∈ {1, . . . , n}}, so the distance between any two distinct vertices is 2, which implies that γ (C * n , r) = 1 for any r > 1 and γ (C * n , r) 2n for r 1. The case r = 1 2 is related to the topic of kissing numbers. Let K ⊂ R n be a convex body, that is, both compact and convex. The (translative) kissing number k (K) is the maximum number of translates of K such that no two translates overlap each other and each translate touches but does not overlap with K. In the case of 1 2 C * n , we have is a packing set for 1 2 C * n and and since k (K) is invariant under the scaling of K, k (C * n ) + 1 = k For the cross-polytope, it is known that k (C * 3 ) = 18 [13,18]. This result implies that γ C * 3 , 1 2 19, however, due to the requirement that one point is the origin, it does not a priori provide an upper bound for any packing set for 1 2 C * 3 . An upper bound for the kissing number of any convex body K is obtained from a result of Hadwiger [11], where this inequality is an equality iff K is a parallelepiped. The cross-polytope is not a parallelepiped, so k (C * 3 ) 25, which results in an upper bound of γ C * 3 , 1 We will work with values of r only in the interval 1 2 , 1 unless otherwise stated. For dimension n and r ∈ 1 − 1 n , 1 , 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. Theorem 1. Let n 2, then γ (C * n , r) = 2n for any r ∈ 1 − 1 n , 1 . Additionally, 1 − 1 n , 1 is the largest possible interval such that γ (C * n , r) = 2n for all r in the interval.
For the n-dimensional ball B n , Hajós and Davenport showed that γ (B n , r) n + 1 for r > 1 √ 2 , as noted by [6]. A closely related topic is the problem of packing a set with copies of another set. This problem has been explored mainly in dimension 2. Let K and L be origin-symmetric convex sets with nonempty interior, then let  The compendium of Goodman, O'Rourke, and Tóth [8] lists known quantities of M (L, B 2 , m) for when L is a square, a circle, and an equilateral triangle and various values of m, usually small. In three dimensions, M (L, B 3 , m) is known for small m and when L is a cube, a cross-polytope, and a tetrahedron [8]. A related problem of packings of squares and rectangles in squares is described in [4]. Let B n be the unit Euclidean ball in n dimensions. Böröczky Jr. and Wintsche have obtained M (C * n , B n , m) for n 3 and m = {3, . . . , 2n + 1} [1].
Let K and B be convex bodies, s > 0, and let D (s, K, B) be a packing set of C * n such that |{x ∈ D (s, K, B) | K + x ⊆ sB}| is maximal among all packing sets of C * n . The density of the densest packing of K, or the packing density of K, is defined to be see Definition 4 in Section 20 of [10] (page 225), and it is independent of B. Then we can set B = C * n and also suppose that K = C * n . For s > 1, since Next, scale this set by a factor of 1 s−1 to get It follows from the definition of the packing density that Hence the packing density of C * n is related to γ (C * n , r) in the sense that γ (C * n , r) 1 − 1 1+r n ∼ δ (C * n ) as r → 0. We now mention some related results involving circle packings in a circle and sphere packings in a cylinder. For the problem of sphere packing inside a cylinder of fixed width in three dimensions, Fu et al. [7] predict that as the radius of the spheres approach zero, densest packings resemble the face-centered cubic lattice-a densest sphere packing in three dimensions [2]-except for the spheres that are near the walls of the cylinder. In the case of dimension two the densest circle packing is generated by the hexagonal lattice [2]. Hopkins, Stillinger, and Torquato [12] provide examples of this phenomenon for dense packings of circles inside a large circle under the condition that the large circle has the same center as one of the small circles. Schürmann [15,16] has shown that under certain conditions the best finite packings of strictly convex bodies can only be obtained using nonlattice packings. Other dense arrangements of k circles within a large circle include modified wedge hexagonal packings and curved hexagonal packings [12], which are the best known packings for some values of k [14].
The density of an infinite packing of a convex set is the fraction of space taken up by congruent copies of the set. Fejes Tóth, Fodor, and Vígh [5] describe some upper bounds for the packing density of C * n , including an asymptotic δ (C * n ) C · 0.86850 n for n 7 and a fixed constant C. Furthermore, they conjecture that δ (C * n ) C · 0.82886 n . Basic facts about convexity and the cross-polytope can be found in books such as the ones from Gruber [9], Ziegler [20], and Coxeter [3], and about packings are in Conway and Sloane [2], Gruber [9], and Zong [21]. Additional details on the kissing number are also in Zong [21]. Section 2 of this paper provides the notation and preliminaries that will be used for the rest of the text. Section 3 contains the proof of the n-dimensional case, Theorem 1. Section 4 proves the equalities and upper bound present in the three parts of the the electronic journal of combinatorics 27(3) (2020), #3.38 3-dimensional case, Theorem 2, introducing additional notation as needed. Theorem 3 is proved in Section 5, and finally Section 6 presents a gallery of diagrams related to these lower bounds.

General notation and preliminaries
Here we introduce notation that will be used over the course of this paper. For a given r > 0, let P n (r) ⊂ C * n be a packing set of rC * n . For any polytope K, let vert (K) be the set of its vertices. For a fixed n ∈ N, define sets V n and S n (r) as follows: . . , n}} and S n (r) := (V n + 2r int (C * n )) ∩ C * n . Therefore S n (r) is the set of all points in C * n that are of distance < 2r from some vertex of C * n . For p ∈ R n and r > 0, we use the notation to denote the interior of the cross-polytope centered at p and scaled by the factor r.
The following lemma is necessary for the general n-dimensional case.
where X is the closure of X, and similarly for −e j instead of e j .
Proof. Without loss of generality we take the e j case. Let y = n i=1 y i e i ∈ C * n ∩C (e j , 2r), then ||y|| 1 1 and ||y − e j || 1 2r. Then the distance from y to (1 − r) e j is Similarly, if y j + r − 1 < 0 then since ||y − e j || 1 2r, 3 Proof of Theorem 1: the n-dimensional case In this section we assume that n 2. We will show that for any r ∈ 1 − 1 n , 1 and any packing set P n (r), the number of points in P n (r) ∩ S n (r) is bounded above by the number of vertices of C * n . Then |P n (r)| 2n and this inequality is true for all P n (r), so γ (C * n , r) 2n. As mentioned in the introduction, the set of vertices V n ⊂ C * n is a packing set of rC * n , which means that 2n is also a lower bound, and so γ (C * n , r) = 2n.
Proof. By definition, S n (r) ⊆ C * n , and so it remains to show the reverse inclusion. Let x ∈ C * n and without loss of generality it can be assumed that x is in the convex hull of 0, e 1 , . . . , e n . Then So every point in C * n is within distance 2r from some vertex of C * n . The next lemma will be crucial for showing that the number of points in P n (r)∩S n (r) is bounded above by the number of vertices of C * n . It is a uniqueness condition which shows that if a point p ∈ P n (r) ∩ S n (r) is close to a vertex v of C * n , specifically ||p − v|| 1 < 2r, then no other point in P n (r) can be close to v.
n has the property that v ∈ C (p, 2r)∩C (q, 2r) for some p, q ∈ P n (r) ∩ S n (r), then p = q.
Proof. Without loss of generality, let v = e j for some j ∈ {1, . . . , n}, then by hypothesis e j ∈ C (p, 2r)∩C (q, 2r). It suffices to show that ||p − q|| 1 < 2r since the distance between two distinct points in P n,r must be 2r or greater.. Then in turn, p, q ∈ C (e j , 2r). Since 2r) is open there exists a r < r (r depends on p and q) such that p, q ∈ C (e j , 2r ). Then it follows from Lemma 4 applied to C * n ∩ C (e j , 2r ) that The following lemma will be used both here and in the 3-dimensional cases in the next section.
(this set may be empty) and a map f : First we need to show that f is well-defined. Let p, q ∈ P n (r) ∩ S n (r) be points such that v ∈ C (p, 2r) ∩ C (q, 2r) for some v ∈ V n , then p = q by Lemma 6, which justifies the use of the words "the point" in the definition of f . From the definition of S n (r), every point p ∈ P n (r) ∩ S n (r) has the property that there is some v ∈ V n such that ||p − v|| 1 < 2r, so f is surjective. Both the domain and range of f are finite sets, so the cardinality of the range can be bounded above by completing the proof. Now we prove Theorem 1. With the preparation above, the proof is mostly a matter of putting together earlier lemmas.
Proof of Theorem 1. Assume that P n (r) is nonempty, otherwise |P n (r)| = 0 and there is nothing to prove. Since r ∈ 1 − 1 n , 1 , it follows from Lemma 5 that C * n = S n , so |P n (r) ∩ S n (r)| is nonempty. Then Lemma 7 shows that |P n (r)| = |P n (r) ∩ S n (r)| 2n. This inequality holds for any P n (r), so γ (C * n , r) 2n for n 2 and r ∈ 1 − 1 n , 1 .
The interval r ∈ 1 − 1 n , 1 cannot be extended in either direction, because γ (C * n , r) = 1 for r > 1 and in Proposition 15 we construct a packing set of rC * n , for r 1 − 1 n , with 2n + 2 points in C * n . For such r, S n (r) C * n and specifically the centroid of each facet is not in S n (r) (cf. Subsection 4.1), so the set consisting of the 2n vertices of C * n and the two centroids on opposing facets of C * n is a packing set of rC * n . Therefore, r ∈ 1 − 1 n , 1 is the largest possible interval such that γ (C * n , r) = 2n is true.

Proof of Theorem 2 (the 3-dimensional case)
When r 2 3 the set S 3 (r) no longer covers all of C * 3 , so unlike the n-dimensional case above, the proofs for the three-dimensional cases require consideration of the remainder C * 3 \S 3 (r).
The endpoints of the range r ∈ 1 2 , 2 3 are 2 3 and 1 2 . For r = 2 3 the set reduces to the centroid of the facet conv {e 1 , e 2 , e 3 }, and when r = 1 2 the set is which contains the midpoints of the edges of the facet conv {e 1 , e 2 , e 3 }. Subsets defined using midpoints of edges are used to solve the related problems of finding upper bounds for k (C * 3 ) and M (C * n , B n , m) for some values of n and m. To find the kissing number of the cross-polytope, Larman and Zong [13] divided the boundary of the cross-polytope into the union of 18 subsets including sets of the form where m is a midpoint of an edge in V 3 , and showed that each subset could contain the center of at most one cross-polytope, resulting in k (C * 3 ) 18. Another method to prove that k (C * 3 ) 18 was used by Talata [18], who showed that any packing set achieving a kissing number of 18 must consist of six points on the vertices, six points on the midpoints of the edges of two opposing facets, and the remaining points on the hexagon passing through the midpoints of the other edges. Böröczky and Wintsche [1] use sets defined by vertices and midpoints of edges to determine an upper bound for M (C * n , B n , m) where n 3 and m ∈ {4, . . . , 2n}.
We will use an approach that has similarities to Larman and Zong [13] and Böröczky and Wintsche [1] in that the maximum distance between any two points in conv (V (r, (σ 1 , σ 2 , σ 3 ))) is less than 2r. Then each conv (V (r, (σ 1 , σ 2 , σ 3 ))) can contain at most one point of P 3 (r), and the number of points of P 3 (r) in S * 3 \S 3 (r) is bounded above by 8 (see Subsection 4.4).
From Proposition 16 below, there is a 10-point packing set for rC * 3 contained in C * 3 , so in fact γ (C * 3 , r) = 10.
Using the above lemma and the same argument as after Lemma 10 in the last subsection, we have |P 3 (r) ∩ (C * 3 \S 3 (r))| 7. However, as in the previous subsection it is possible to lower the 7 to a 6 with the following argument.
The proof of Theorem 2 (b) is virtually identical to the proof of Theorem 2 (a).

Constructive lower bounds including the proof of Prop. 3
In contrast to the upper bounds, the lower bounds are all obtained by explicit constructions of points in the cross-polytope. For n = 3 and r ∈ 1 2 , 2 3 , all of the constructions shown here contain the six points of V 3 and the remaining points are in the union of the eight sets conv (V (r, (σ 1 , σ 2 , σ 3 ))). There are no claims of uniqueness made here; more than one set of points may achieve the lower bounds of Theorems 2 and 3. The proofs of these propositions amount to checking that the distance between any two distinct points in the packing set is at least 2r, and the calculations can be performed by hand or using a computer.

Figure 4:
The green cross-polytopes represent the sets C x, 2 3 for x ∈ V 3 and the blue cross-polytopes represent the sets C y, 2 3 for y ∈ Q 10 .